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DIFFRACTION STUDIES OF THE ATOMICSTRUCTURE OF GRAIN BOUNDARIES

J. Budai, S. Sass

To cite this version:J. Budai, S. Sass. DIFFRACTION STUDIES OF THE ATOMIC STRUCTURE OF GRAIN BOUND-ARIES. Journal de Physique Colloques, 1982, 43 (C6), pp.C6-103-C6-113. �10.1051/jphyscol:1982611�.�jpa-00222291�

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Colloque C6, supplément au n° 12, Tome 43, dêaembre 1982 page C6-103

Résumé. - Une revue des récents résultats obtenus sur la structure atomique des joints de grain par des techniques de diffraction est présentée. Pour la première fois, la structure d'un joint de grain est déterminée seulement à partir des données de diffraction. L' influence d'un métal particulier sur la structure de joint est aussi considérée. Par ailleurs, la variation de l'épaisseur de joint de grain est établie en fonction de paramètres tels que l'angle de désorientation et l'orientation du plan de joint. Grâce aux résultats d'études'de diffraction, des généralisations sont avancées sur la structure des joints de forte désorientation.

DIFFRACTION STUDIES OF THE ATOMIC STRUCTURE OF GRAIN BOUNDARIES

J . Budai and S.L. Sass

Department of Materials Soienoe and Engineering, Cornell University, Ithaca, NY 14853, U.S.A.

Abstract. - The recent results of the application of diffraction techniques to study the atomic structure of grain boundaries are reviewed. The outcome of the first determination of a grain boundary structure using diffraction information alone is examined. The influence of a particular metal on the grain boundary structure is examined. The variation of grain boundary structural width with parameters such as misorientation angle and boundary plane is studied. Generalizations are made concerning the structure of large angle boundaries based on the results of the diffraction studies.

I. Introduction There have been a large number of studies made using sophisticated

experimental techniques to examine the structure of grain boundaries. Transmission electron microscopy studies have revealed the dislocation content of various boundaries (see, for example, Ref. 1) and the rigid body translation between the two crystals neighboring the boundary (2). For tilt boundaries, high resolution microscopy techniques have provided information with resolution approaching atomic dimensions (3,4). Electron and X-ray diffraction studies have demonstrated through observations in reciprocal space that the boundary is periodic (5), that well defined patterns of relaxation exist (6) and that the boundary structural width can be measured (7). To date, however, none of these experimental techniques have proved capable of determining, without ambiguity, the exact location of all atoms in the grain boundary region. On the theoretical side, sophisticated computer modelling techniques have been developed which can determine the detailed atomic structure of boundaries but which also suffer from a number of limitations. Most notable of these is the choice of a realistic interatomic potential.

As a step towards obtaining a believeable atomic structure, several studies have attempted to correlate experimental observations with structures generated by computer modelling. This approach has met with a certain degree of success with determinations of translation states (8), projected tilt boundary structures (9) and, most recently, of the complete core structure of a twist boundary (10). In the latter study the intensities of extra grain reflections from a z=13*(e=22.62°) [001] twist boundary in gold measured with X-ray diffraction were compared with the structure factors of computer generated boundary structures. The present paper describes how the diffraction approach can be used to determine a grain boundary structure, presents the first determination of a boundary structure made using only diffraction information (11), compares boundary structures in different metals (12), discusses grain boundary structural width measurements, and then makes generalizations concerning the structure of large angle [001] twist boundaries based on diffraction observations.

* I is the reciprocal of the density of coincidence sites.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982611

C6-104 JOURNAL DE PHYSIQUE

I n o r d e r t o b e t t e r unders tand t h e d e t a i l s o f t h e exper imenta l r e s u l t s , i t i s wo r thwh i l e t o examine t h e r e c i p r o c a l l a t t i c e assoc ia ted w i t h t h e g r a i n boundary. A l l o f t h e X-ray obse rva t i ons a r e made on manufactured b i c r y s t a l s (see re fe rence 13 f o r d e t a i l s ) , one o f which i s shown schemat i ca l l y w i t h a smal l ang le t w i s t boundary a t i t s midplane, i n F ig . 1 (a) . The 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 schemat i ca l l y i n F ig . 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 -ax i s 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 assoc ia ted w i t h t h e g r a i n boundary g i ves r i s e t o e x t r a r e f l e c t i o n s , wh ich a r e shown i n t h e f o r m 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 ) s i n c e t h e boundary i s n o t p e r i o d i c a long t h e z - d i r e c t i o n . 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 p lane 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 on to t h e boundary p lane w h i l e t h e l e n g t h o f t h e r e l r o d s i s r e l a t e d t o t h e boundary w id th . The s t r u c t u r a l i n f o r m a t i o n d iscussed i n t h i s paper comes f r o m measurements made on e i t h e r t h e s t r u c t u r e f a c t o r s i n t h e L=O p lane (FHKO) o r t h e r e l r o d p r o f i l e s a long t h e L d i r e c t i o n .

F igu re 1. ( a ) B i c r y s t a l c o n t a i n i n g a smal l ang le [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 a t i t s midplane.

(b ) Schematic th ree-d imens iona l 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 ) .

11. The P r o j e c t e d S t r u c t u r e o f t h e 1=5(8=36.87~) [001] T w i s t Boundary A. The Symmetry o f t h e U n i t C e l l When de te rm in ing t h e s t r u c t u r e o f a p e r i o d i c o b j e c t t h e f i r s t s t e p i s t o

determine t h e dimensions o f t h e u n i t c e l l . I n g r a i n boundar ies s t u d i e s performed on manufactured b i c r y s t a l specimens, t h e u n i t c e l l i s expected t o be f i x e d by t h e cho i ce o f t h e m i s o r i e n t a t i o n a x i s and angle, and t h e boundary plane. The second s t e p i s t o determine t h e symmetry elements o f t h e u n i t c e l l . F ig . 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 1=5 t w i s t boundary, w i t h 2 a tomic p lanes 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 co inc idence s i t e l a t t i c e (CSL) c o n f i g u r a t i o n ; t h a t i s , t h e two c r y s t a l s a r e n o t t r a n s l a t e d away f rom t h e co inc idence p o s i t i o n f o r f .c.c. s t r u c t u r e s . F o l l o w i n g B r i s towe and Crocker (14) , 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 ) . Any t r a n s l a t i o n o f one c r y s t a l w i t h respec t t o t h e o the r , which i s n o t a v e c t o r o f t h e DSC l a t t i c e (15), w i l l change t h e symmetry assoc ia ted w i t h t h e boundary s t r u c t u r e . F ig . 2(d,e) shows t h e symmetry elem n t s o f two h i g h symmetr t r a n s l a t e d s t a t e s c h a r a c t e r i z e d by DSC t r a n s l a t i o n s h / 2 and 61+62/2 (where 81 and 62 a r e t h e or thogona l DSC b a s i s v e c t o r s i n t h e p lane o f t h e boundary (14) ) . Other t r a n s l a t i o n s a r e poss ib le , and f o r these cases, t h e number o f symmetry elements decreases. The symmetry o f t h e u n i t c e l l can be determined f r o m 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 r u l e s . The s t r u c t u r e f a c t o r r u l e s f o r t h e t h r e e t r a n s l a t i o n s t a t e s shown i n F i g 2(c,d,e) a r e g i ven i n t h e Table. A c o m p l i c a t i o n i n t h i s a n a l y s i s a r i s e s f o r t h e 61/2 t r a n s l a t i o n s t a t e , where t h e a c t u a l boundary s t r u c t u r e c o u l d c o n s i s t o f a m i x t u r e o f domains w i t h t h e same t r a n s l a t i o n s t a t e , b u t d i f f e r e n t o r i e n t a t i o n s . For

xample, i n d i f f e r e n t areas o f t h e boundary t h e t r a n s l a t i o n can be e i t h e r 61/2 o r 92/2, y i e l d i n g two domains. I t i t h i s case t h e s c a t t e r i n g f r o m t h e two domains w i l l average t o g e t h e r i n such a way t h a t t h e r e a r e no l o n g e r s t r u c t u r e f a c t o r f o r b i d d e n r e f l e c t i o n s and more s y m e t r y i s shown i n t h e d i f f r a c t i o n p a t t e r n t han a c t u a l l y e x i s t s i n any p a r t i c u l a r domain.

7&:, Unrelaxed 1 5

m - m * Ave [200]

la1 lbl

F igu re 2. ( a ) Schematic diagram o f t h e unre laxed z.5 p r i m i t i v e CSL u n i t c e l l p r o j e c t e d on to t h e g r a i n boundary plane. ( b ) S ide v iew o f t h e same u n i t c e l l w i t h two lanes above and below t h e boundary shown. The symmetry elements o f ( c ) CSL, ( d ) B1/2 t r a n s l a t i o n and ( e ) ( D l + b ) / ~ t r a n s l a t i o n [001] t w i s t boundar ies i n f c c c r y s t a l s . The squares and e l l i p s e s rep resen t f o u r - f o l d and t w o - f o l d axes pe rpend i cu la r t o t h e i n t e r f a c e ; t h e con t i nuous l i n e s rep resen t t w o - f o l d and t w o - f o l d screw axes i n t h e i n t e r f a c e , t h e l a t t e r be ing d i s t i n g u i s h e d by h a l f arrowheads. The diagrams a r e v a l i d f o r a l l va lues o f Z.

Table: D i f f r a c t i o n symmetry and s t r u c t u r e f a c t o r s e l e c t i o n r u l e s

CSL S t r u c t u r e

IFHKLI = I F i i i I =A l F i K L l = lFH iL l = lFHK i l = lFKHLl FHOO = 0 i f H i s odd

FOKO = 0 i f K i s odd

6112 S t r u c t u r e

1 FHKL 1 ' I FKCI = I F ~ K L I = I FHKL I = 1 FHKI I # 1 FKHL 1 FOKO = 0 i f K i s odd

(61 + 62 ) /2 S t r u c t u r e

I F I = I F - - - ] = IF- I = I F - ] = I F - 1 = lFKHLl HKL HKL H KL HKL HKL

No s t r u c t u r e f a c t o r f o rb idden r e f l e c t i o n s .

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The d i s t i n g u i s h i n g d i f fe rence between the CSL and a l l o ther conf igurat ions l i e s i n those s t r u c t u r e f a c t o r se lec t ion r u l e s which are r e l a t e d t o the presence o f two- fo ld screw axes. F r he CSL s t ruc tu re , F (H=odd) and FOK0(K=odd) must be

ero, whereas f o r the ( i 1+ 8 2)/2 t r a n s l a t i o n orHoOfor a mixture o f domains w i t h 6112 and a212 t rans la t ions , i n t e n s i t y can be present a t these posi t ions. The observat ions o f Budai, Bristowe and Sass (11) are consis tent w i t h the c=5 u n i t c e l l being i n the CSL con f igu ra t ion (unt rans lated s ta te ) . The observat ions o f Gaudig and Sass (13) (which were repeated i n more d e t a i l i n r e f . 11) are consis tent w i t h the ~=13(8=22.62") [001] t w i s t boundary a1 so being i n the unt rans lated s tate.

B. The Number o f Independent Atoms i n the U n i t C e l l Once the symmetry has been establ ished the next s tep i s t o determine the

minimum number o f atoms whose pos i t i ons must be loca ted i n order t o completely character ize the u n i t c e l l contents. By examination o f Fig. 2(a) i t i s seen t h a t there are 5 atoms per plane, w i t h the x,y pos i t i ons o f one atom per plane ( a t e i t h e r 1/2, 112 o r 0,O) f i x e d by the t r a n s l a t i o n s t a t e o f the boundary. The remaining f o u r atoms i n the plane are symmetry r e l a t e d by the 4 - f o l d ax is . Thus i t i s necessary t o loca te 2 atoms per plane i n order t o loca te a l l the atoms i n the plane. I f only t h e x,y pos i t i ons are t o be determined, i t i s necessary t o l o c a t e 1 atom per plane i n order t o loca te a l l the atoms i n the plane. I t i s a lso seen t h a t the atoms i n the upper and lower c r y s t a l s are r e l a t e d by the two- fo ld screw axes l y i n g i n the i n t e r f a c e plane. What t h i s means i s , i f i t i s assumed t h a t the g r a i n boundary i s 4 atomic layers t h i c k , then i n order t o determine the atomic s t r u c t u r e pro jected onto the boundary plane, i t i s necessary t o determine the x,y coordinates o f on ly 2 atoms (marked A,B i n Fig. 2 (a ) ) . Most o f the experimental observat ions are i n the L=O plane (see Fig. 1 ( a ) ) , which contains in format ion on ly on the x,y coordinates. Thus the study here w i l l be concerned w i t h determining t h e p ro jec ted boundary s t ruc tu re .

The assumption t h a t s i g n i f i c a n t atomic re laxa t ions f o r a c = 5 boundary are l i m i t e d t o a reg ion which i s on ly 4 atomic layers t h i c k (Q.8nm f o r Au) i s based on both experimental and t h e o r e t i c a l considerat ions. The w id th o f the s t ra ined region f o r a Z = 377 ( 8 = 23.8") t w i s t boundary i n go ld was exper imenta l ly determined, from the i n t e n s i t y p r o f i l e s o f various re l rods , t o be Q.8nm (16). 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 a Z = 5 (8 = 36.9") boundary were q u a l i t a t i v e l y observed t o be broader, imply ing a narrower w id th f o r t h e s t r a i n e d region. For the 8=23.8O case, the boundary thickness was i n good agreement w i t h the 0 - l a t t i c e spacing, d ~ , o f 0.7 nm. This agreement i s consis tent w i t h St. Venant's p r i n c i p l e (17). For the E = 5 boundary, the 0 - l a t t i c e spacing i s on ly 0.456 nm and a narrower boundary reg ion i s expected. I f the stresses decrease exponent ia l ly as exp (-ZIT z/dD) (17), then the displacements a t a distance, z, corresponding t o the t h i r d atomic plane from the boundary, would be a f a c t o r o f 275 times smal ler than the displacements i n the f i r s t plane. Thus the assumption t h a t s i g n i f i c a n t re laxa t ions are r e s t r i c t e d t o a fou r - layer reg ion i n the v i c i n i t y o f t h e boundary plane i s q u i t e reasonable.

C. The Atomic Posi t ions i n the U n i t Cel l . Fo l lowing upon the conclusions o f the previous sect ions, the determinat ion o f

the pro jected 1=5 boundary s t r u c t u r e proceeds by the 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 determinat ions (18). I n t h i s method the x and y coordinates f o r the two independent atoms l a b e l l e d A and B i n Fig. 2(a) are scanned i n small increments over an area covering the range o f atomic displacements which generate a l l poss ib le g r a i n boundary conf igurat ions. For each con f igu ra t ion the magnitude o f the s t r u c t u r e fac to rs are ca lcu la ted and compared t o exper imenta l ly observed s t ruc tu re f a c t o r s us ing the d e f i n i t i o n o f the r e l i a b i l i t y f a c t o r , R, given below,

where F ? ~ = the observed s t r u c t u r e f a c t o r o f the j t h r e f l e c t i o n . J

Fcal j

= 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 .

W . = w e i g h t i n g f a c t o r (o<_W.<l) 3 J -

The s e t o f XA, y ~ , xg, y g coo rd ina tes wh ich l eads t o t h e s m a l l e s t va lue f o r R, p rov ides t h e b e s t f i t t o t h e d i f f r a c t i o n observat ions . Thus t h e de te rm ina t i on o f t h e 1 = 5 s t r u c t u r e becomes a search f o r minimum va lues f o r t h e f u n c t i o n R(xA, y ~ , XB, yg) . Such a search y i e l d s t h e 2=5 s t r u c t u r e shown i n F ig . 3.

q + - - - - - - - - - - - J Q I Q I 'a d

b I +

I I

I

I I 4B l a I

A1 Q-i I

d I 8 - - - - - - - - - - - q!J

F igu re 3. Atomic displacements f o r t h e p r o j e c t e d C=5 CSL u n i t c e l l . T h i s s t r u c t u r e had

t h e s m a l l e s t R(=0.15).

Examinat ion o f F ig . 3 revea l s seve ra l i n t e r e s t i n g p o i n t s about t h e s t r u c t u r e . Note t h a t t h e displacement assoc ia ted w i t h atom A ( f i r s t p lane f rom t h e boundary) i s much l a r g e r than t h e displacement assoc ia ted w i t h atom 6 (second p lane f rom boundary). T h i s r a p i d decrease i n magnitude f o r t h e a tomic r e l a x a t i o n s i n p lanes away f rom t h e boundary i s as expected f o r a l a r g e ang le boundary. I n o r d e r t o unders tand t h e o r i g i n o f t h e r e l a x a t i o n s , cons ide r t h e f o r c e s a c t i n g on atom A. The atoms i n t h e lower c r y s t a l a c t t o h o l d atom A i n i t s unre laxed p o s i t i o n i n o r d e r t o preserve f .c.c. s tack ing. The atoms i n t h e upper c r y s t a l a c t t o d i s p l a c e atom A, w i t h t h e nea res t atoms hav ing t h e l a r g e s t e f f e c t . I n t h e unre laxed c o n f i g u r a t i o n , atoms A and A1 a r e 23% c l o s e r than t h e nea res t ne ighbor d i s tance i n a p e r f e c t f .c .c . c r y s t a l . Thus t h e obse rva t i on t h a t i n t h i s s t r u c t u r e these atoms move a p a r t i s p h y s i c a l l y q u i t e reasonable. Examinat ion o f 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 r e l a t e d displacements wh ich can be i n t e r p r e t e d as l o c a l r o t a t i o n s about ' 0 ' elements. For [OOl] t w i s t boundar ies, t h e ' 0 ' e lements c o n s i s t o f rods l y i n g pe rpend i cu la r t o t h e boundary, some o f wh ich pass through CSL s i t e s (15) . The degree o f r o t a t i o n i s l a r g e ( ~ 2 0 " ) .

A t t h i s p o i n t , i t i s impor tan t t o a t t emp t t o unders tand t h e c=5 boundary s t r u c t u r e i n terms o f more genera l p h y s i c a l concepts wh ich may be a p p l i c a b l e t o o t h e r boundar ies. The e x i s t e n c e o f r o t a t i o n a l t y p e r e l a x a t i o n s i n g r a i n boundar ies was f i r s t recogn ized i n connect ion w i t h l ow ang le t w i s t boundar ies (1 ) . Here, t h e observed s t r u c t u r e can be i n t e r p r e t e d as l o c a l r o t a t i o n s about ' 0 ' p o i n t s o f approx imate ly 6 /2 which produced l a r g e areas o f nea r p e r f e c t l a t t i c e separa ted by narrow reg ions o f m i s f i t ( l a t t i c e screw d i s l o c a t i o n s ) . C l e a r l y , i f l o c a l r o t a t i o n s o f about 9/2 a r e t o occur i n l a r g e ang le boundar ies then l a r g e a tomic displacements would be i nvo l ved . Such displacements were n o t found t o occur i n t h e ~ = 5 computer s i m u l a t i o n s (11) b u t were a r e s u l t o f t h e d i f f r a c t i o n a n a l y s i s , as can be seen i n F ig . 3.

F u r t h e r i n s i g h t i n t o t h e b n i q u e n a t u r e o f t h e r e l a x a t i o n s assoc ia ted w i t h t he s t r u c t u r e generated by t h e d i f f r a c t i o n a n a l y s i s can be ga ined by examining F i g . 4 (a ) wh ich shows s i x unre laxed u n i t c e l l s c o n t a i n i n g o n l y t h e atoms i n t h e f i r s t p lanes above and below t h e boundary. As ment ioned e a r l i e r , t h e r e l a x a t i o n of any "wh i t e " atom depends on a ba lance between t h e r e p u l s i v e f o r c e s o f t h e nearby "b lack " atoms and t h e r e s t o r i n g f o r c e o f t h e "wh i t e " s i n g l e c r y s t a l . F igu re 4 ( b ) shows t h e s t r u c t u r e determined by t h e d i f f r a c t i o n ana l ys i s . The "wh i te " atoms a r e l o c a t e d approx imate ly on t h e edges o f t h e CSL u n i t c e l l , p o s i t i o n s o f spec ia l symmetry. It can be seen t h a t t h e magnitudes o f t h e r e l a x a t i o n s a r e such as t o maximize t h e d i s t a n c e between t h e "wh i t e " and "b lack " atoms.

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0 ' 0 ' 0 ' I . 8 A I .

i 0 8 0 I 0 . _ _ . _ . _ _ _ . _ _ . . . _ _ _ I . ^ _ _ _ _ . . I

Unrelaxed

.-. - . . - - .. - 0 - - - - . A - - * - - . . -. Relaxed

I Unrelaxed Unrelaxed Lower Crystal G u P p e r crystal

-- ---A--- ---

I

I I

lbl

Figure 4. ( a ) Projected unrelaxed c=5 CSL uni t c e l l containing only the atoms i n the f i r s t planes above and below the grain boundary. ( b ) Projected relaxed c=5 CSL uni t c e l l determined by the d i f f r a c t i o n analysis . ( c ) I l l u s t r a t i n g t h e atomic configuration i n t h e v i c i n i t y of a un i t c e l l corner.

I n Fig. 4(c) the s i t u a t i o n i n the v i c i n i t y o f a u n i t c e l l corner i n Fig. 4(a,b) i s shown bu t now w i t h the 2nd, 4th, 6th, etc., planes above the boundary ind ica ted by a b lack c i r c l e e a n d the l s t , 3rd, 5th, etc., planes below the boundary ind ica ted by a whi te diamondo. Here i t i s seen t h a t the o atom i n the 1 s t plane below the boundary has ro ta ted by ~ 8 / 2 ( i n d i c a t e d by the arrow) i n a clockwise sense t o a p o s i t i o n which i s approximately on the u n i t c e l l edge. I t i s c l e a r t h a t the displaced 0 atom i s i n the p o s i t i o n o f the median f c c u n i t c e l l between the upper and lower c rys ta ls . Thus the d i f f r a c t i o n s t r u c t u r e conforms t o the concept, o r i g i n a l l y recognized i n low angle boundaries, t h a t def ines the i n t e r f a c e as discontiguous patches o f median f c c s t ruc tu re . It i s i n t e r e s t i n g t h a t examination o f the s t ruc tu re o f the z=13 boundary determined by Bristowe and Sass (10) [Fig. 4(c) o f t h e i r paper] shows the presence o f the same l o c a l s t ruc tu re , s ince atoms above and below the boundary plane r o t a t e about ' 0 ' elements by an angle o f about 012. As a l o g i c a l extension o f these r e s u l t s , i t i s tempting t o genera l ize the presence o f patches o f median f c c s t r u c t u r e t o a l l h igh angle [ O O l ] t w i s t boundaries.

111. Comparison o f the S t ruc tu re o f Twist Grain Boundaries i n Au and Ag (12) I n a recent computer model1 i n g study (10) the atomic s t r u c t u r e o f the

e=22.62" (c=13) [001] t w i s t boundary was ca lcu la ted f o r several f.c.c. metals. It was p red ic ted t h a t the s t r u c t u r e o f t h i s type o f boundary should be d i f f e r e n t i n d i f f e r e n t metals. Since the p roper t ies o f a boundary are expected t o be dependent i n some manner on the d e t a i l s o f i t s s t ruc tu re , i t i s p a r t i c u l a r l y important t o check the v a l i d i t y o f t h i s p red ic t ion . This was done by comparing the s t r u c t u r e o f the z=13 [001] t w i s t boundary i n Au and Ag using X-ray d i f f r a c t i o n techniques. 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 pa t te rns showing the same regions of rec ip roca l space were obta ined and compared f o r Au and Ag b i c r y s t a l specimens which con ta in the ~ = 1 3 t w i s t boundary. The i n t e n s i t i e s o f the r e f l e c t i o n s i n these pa t te rns a re r e l a t e d t o the square o f the s t r u c t u r e f a c t o r s o f these r e f l e c t i o n s . I t i s expected t h a t i f the boundary s t ruc tu res are s i g n i f i c a n t l y d i f f e r e n t , then the r e l a t i v e i n t e n s i t i e s o f the g r a i n boundary r e f l e c t i o n s observed from the Au and Ag boundaries w i l l a1 so be d i f f e r e n t .

Since the l a t t i c e parameters o f Au and Ag are approximately the same (ao = 0.40783nm f o r Au and ao = 0.40856nm f o r Ag), the d i f f r a c t i o n geometry i s the same f o r both metals. It i s poss ib le then t o search f o r any d i f fe rences i n the s t r u c t u r e f a c t o r by simply examining the same type d i f f r a c t i o n pa t te rns from the Ag and Au specimens and look ing f o r d i f fe rences i n the r e l a t i v e i n t e n s i t i e s o f p a r t i c u l a r g r a i n boundary r e f l e c t i o n s .

Examples o f the experimental observat ions are presented i n the form o f p a i r s o f X-ray d i f f r a c t i o n pat terns taken i n the v i c i n i t y o f the f.c.c. r e f l e c t i o n s i n Fig. l ( b ) w i t h the same d i f f r a c t i o n geometry f o r the Au and Ag g r a i n boundaries. Fig. 5(a,b) shows pa t te rns f o r Au and Ag, respect ive ly , taken i n the v i c i n i t y o f the 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 the Ewald sphere. The ind ices o f the g r a i n boundary r e f l e c t i o n s are ind ica ted on the pa t te rns i n Fig. 5(a,b) us ing the H,K n o t a t i o n f o r the CSL u n i t c e l l . C lear l y present i n the Au pa t te rn are several g r a i n boundary r e f l e c t i o n s . These can be compared t o the pa t te rn from Ag i n Fig. 5(b) and i t i s seen t h a t the r e l a t i v e i n t e n s i t i e s o f the r e f l e c t i o n s i n the two pat terns are q u i t e s i m i l a r . For example, 1 i s t i n g the observed r e f 1 ect ions i n decreasing order o f i n t e n s i t y y i e l d s the sequence 6,6; 4,4; 2,6; 3,5 f o r both Au and Ag.

Fiq. 5(c,d) shows pa t te rns f o r Au and Ag, respec t i ve ly , taken i n the v i c i n i t y o f the 220 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 s 11,1,0 and 11,1,0 on the Ewald sphere. Again, the 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 ra in boundary r e f l e c t i o n s i n both pat terns are q u i t e s i m i l a r . For example, i t i s seen t h a t the r e f l e c t i o n s 10,O; 10,4; 11,l are i n a decreasing order o f i n t e n s i t y f o r both Au and Ag -

Fig. 5(e,f) shows pa t te rns f o r Au and Ag, respect ive ly , taken i n the v i c i n i t y o f the 400 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 10,10,0 on the Ewald sphere. Again i t i s seen t h a t the r e l a t i v e i n t e n s i t i e s o f the g r a i n boundary r e f l e c t i o n s are approximately the same i n both the Au and Ag pat terns.

As p a r t o f t h i s study, the e f f e c t o f small changes i n the boundary s t r u c t u r e on the ca lcu la ted s t r u c t u r e f a c t o r s was examined. It was found t h a t two s t ruc tu res

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Figure 5. X-ray d i f f r a c t i o n pat terns f o r the c=13 0011 t w i s t boundary taken w i t h 4 hr. exposures us ing synchrotron r I d ia t ion .

(a ) and (b ) The 200 region o f rec ip roca l space w i t h g r a i n boundary r e f l e c t i o n 6,6,0 on the Ewald sphere. Ref lect ions due t (1) twins are ind ica ted

harmonics are ind ica ted by H. (a) Au (b) Ag.

a by T, (2 ) double d i f f r a c t i o n are ind ica ted by D and ( 3 ) wavelength

(c ) and (d ) The 220 region-of rec ip roca l space 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 ,l ,O* on the Ewald sphere. ( c ) Au (d) Ag.

(e ) and ( f ) The 400 region o f rec ip roca l space w i t h g r a i n boundary r e f l e c t i o n 10,10,0 on the Ewald sphere. (e) Au ( f ) Ag.

w i t h a tomic p o s i t i o n s which d i f f e r by o n l y 0.005 nm can g i v e r i s e t o q u i t e d i f f e r e n t s t r u c t u r e f a c t o r s . The weak r e f l e c t i o n s a r e e s p e c i a l l y s e n s i t i v e t o smal l changes i n s t r u c t u r e . Thus i t i s b e l i e v e d t h a t i f two s e t s o f g r a i n boundary s t r u c t u r e f a c t o r obse rva t i ons a r e s i m i l a r , t h e cor respond ing s t r u c t u r e s a r e a l s o s i m i l a r .

What can be s a i d now about t h e s t r u c t u r e s o f t h e Au and Ag boundar ies? The s i m i l a r i t y between 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 f o r t h e Au and Ag boundar ies, i n c l u d i n g obse rva t i ons f o r LfO as w e l l as L=O (12), suggests t h a t t h e Au and Ag boundary s t r u c t u r e s a r e a l s o s i m i l a r . Since most o f t h e obse rva t i ons were made i n t h e L=O plane, t h i s s ta tement i s s t r o n g e s t w i t h r e s p e c t t o t h e s t r u c t u r e s o f t h e Au and Ag ~ = 1 3 [OOl] t w i s t boundary p r o j e c t e d on to t h e boundary plane.

It remains now t o d i scuss why t h e Au and Ag boundary s t r u c t u r e s a r e found e x p e r i m e n t a l l y t o be s i m i l a r , when computer mode l l i ng p r e d i c t s t h a t they shou ld be d i f f e r e n t . I n t h e g r a i n boundary s tudy conducted here, what i s n o t known i s t h e c o n c e n t r a t i o n o f i m p u r i t i e s a t t h e boundary plane. S ince t h e s i n t e r i n g t o g e t h e r o f t h e two s i n g l e c r y s t a l s t o produce t h e b i c r y s t a l was c a r r i e d o u t i n a i r , i t can be expected t h a t e i t h e r a s u l f i d e o r an o x i d e i s p resen t on t h e su r face o f t h e s i l v e r s i n g l e c r y s t a l s . A f t e r s i n t e r i n g , i t i s reasonab le t o expect t h a t t h e r e i s a second component d i s s o l v e d i n t h e Ag i n t h e v i c i n i t y o f t h e boundary. There may a l s o be a smal l amount o f Ag d i s s o l v e d i n t h e Au b i c r y s t a l . Thus t h e comparison be ing made i n t h e s tudy may a c t u a l l y be between a Au boundary c o n t a i n i n g a smal l amount o f Ag and a Ag boundary c o n t a i n i n g a smal l amount o f i m p u r i t y such as oxygen o r s u l f u r . It i s d i f f i c u l t t o b e l i e v e , however, t h a t t h e presence o f i m p u r i t i e s can mod i f y t h e Ag boundary s t r u c t u r e , wh ich i s a c t u a l l y d i f f e r e n t f r o m t h e Au s t r u c t u r e , and cause i t t o be s i m i l a r . T h i s d i f f r a c t i o n s tudy has been r e c e n t l y repeated u s i n g Au and Ag b i c r y s t a l s produced by e p i t a x i a l growth and s i n t e r i n g under UHV c o n d i t i o n s , and t h e obse rva t i ons a r e t h e same as shown i n F ig . 5(19). T h i s r e s u l t i n d i c a t e s t h a t i m p u r i t i e s a r e n o t s t r o n g l y i n f l u e n c i n g t h e obse rva t i ons i n F ig . 5. A t p resen t t h e o r i g i n o f t h e disagreement between t h e r e s u l t s o f t h e X-ray exper iment and t h e computer m o d e l l i n g p r e d i c t i o n i s n o t understood. However, t h e s i m i l a r i t y o f t h e Au and Ag obse rva t i ons suggests t h a t t h e f i n e d e t a i l s o f t h e i n t e r a t o m i c p o t e n t i a l a r e n o t impor tan t i n de te rm in ing t h e boundary s t r u c t u r e .

I V . Measurements o f G ra in Boundary S t r u c t u r a l Wid th(7) An impor tan t s t r u c t u r a l c h a r a c t e r i s t i c o f a g r a i n boundary i s i t s t h i ckness .

When d i scuss ing th i ckness , i t i s necessary t o s p e c i f y what i s meant by t h i s term. For example, i n t h e d i f f u s i o n l i t e r a t u r e , t h e t e r m " th i ckness " r e f e r s t o t h e " d i f f u s i o n w id th " which i s a measure o f t h e w i d t h o f t h e boundary r e g i o n where t h e d i f f u s i n g spec ies i s h i g h l y mobi le. Another p o s s i b l e use o f t h e t e rm r e f e r s t o t h e " s t r u c t u r a l w id th " which i s a measure o f t h e depth o f p e n e t r a t i o n o f t h e g r a i n boundary d isp lacement f i e l d i n t o t h e ne ighbo r i ng s i n g l e c r y s t a l s . It i s expected f rom t h e o r e t i c a l c o n s i d e r a t i o n s t h a t t h e magnitude o f t h e s t r u c t u r a l w i d t h w i l l be r e l a t e d t o t h e magnitude o f t h e p e r i o d i c i t y i n t h e p lane o f t h e boundary (see, f o r example, r e f . 17).

The g r a i n boundary may be represented by a s l a b o f m a t e r i a l c o n t a i n i n g a p e r i o d i c displacement f i e l d wh ich has d i s p l a c e d t h e atoms f rom t h e i r p e r f e c t c r y s t a l p o s i t i o n s i n a c e r t a i n number o f p lanes on b o t h s i des o f t h e boundary plane. Because the'number o f pe r tu rbed p lanes i s smal l , t h e r e f l e c t i o n s assoc ia ted w i t h t h i s r e g i o n a r e e longa ted t o g i v e r e l r o d s a long t h e d i r e c t i o n normal t o t h e boundary plane. I f , f o r example, t h e d isp lacement f i e l d a f f e c t e d o n l y one plane, t h e g r a i n boundary would g i v e r i s e t o a s e t o f i n f i n i t e l y l o n g r e l r o d s which a r e a r rayed a long t h e d i r e c t i o n o f t h e p e r i o d i c i t y . The a c t u a l l e n g t h o f t h e r e l r o d , t he re fo re , p rov ides a measure o f t h e w i d t h o f t h e r e g i o n c o n t a i n i n g t h e displacement f i e l d . By combining t h e r e s u l t s o f X-ray d i f f r a c t i o n (16 ) and computer m o d e l l i n g s t u d i e s ( l o ) , i t was suggested t h a t t h e d i f f r a c t i o n techn ique measures t h e w i d t h o f t h e r e g i o n o f l a r g e s t r a i n a t t h e boundary.

Hagege, Ca r te r , Cosandey and Sass (7) used an e l e c t r o n d i f f r a c t i o n techn ique t o measure t h e l e n g t h o f t h e r e l r o d s f rom edge-on [001] t i lt g r a i n boundar ies con ta ined i n Au b i c r y s t a l s . A t y p i c a l obse rva t i on i s presented i n F ig . 6 where t h e edge-on boundary and i t s assoc ia ted s t r e a k s a re shown. F ig . 7 summarizes a l l t h e exper imenta l measurements made on t h e s t r u c t u r a l w i d t h o f (100) and (110)

C6-112 JOURNAL DE PHYSIQUE

boundar ies. A l so shown i s $he v a r i a t i o n o f dislocation spacing, d ~ , w i t h 8 g i ven by t h e Frank fo rmu la dD f lb1/2sine/2, where b i s t h e Burgers ves tor . (For t h e (100) boundary plane, lbl = ao; f o r t h e (110) boundary plane, ( b l = aod?/2).

lbl F igu re b. e=lOO [ O O l ] (010) g r a i n boundary. ( a ) B r i g h t - f i e l d image showing

a p e r i o d i c a r r a y o f d i s l o c a t i o n s a long t h e [ l oo ] av. d i r e c t i o n . The d i s l o c a t i o n spac ing i s 2.3+0-1 nm. (b ) D i f f r a c t i o n p a t t e r n f o r (a ) . The w h i t e arrows p o i n t t o e longa ted g r a i n boundary r e f l e c t i o n s . Note t h e v e r y c l e a r s t r e a k i n t h e v i c i n i t y o f t h e 220 reg ion .

10 20 30 4 0 MISORIENTATION ANGLE. 8

F i g u r e 7. The exper imenta l measurements o f g r a i n boundary w i d t h p l o t t e d versus t h e m i s o r i e n t a t i o n angle. Open and f i l l e d c i r c l e s correspond t o t h e (100) and (110) boundary planes, r e s p e c t i v e l y . These measurements a r e compared w i t h curves showing t h e v a r i a t i o n o f t h e p e r i o d i c i t y i n t h e boundary p lane c a l c u l a t e d u s i n g t h e Frank formula .

The s t r u c t u r a l w i d t h measurements and t h e va lues o f t h e d i s l o c a t i o n spac ing c o r r e l a t e w e l l , which i s an exper imenta l c o n f i r m a t i o n o f t h e t h e o r e t i c a l p r e d i c t i o n t h a t t h e w i d t h o f t h e s t r a i n e d r e g i o n c r e a t e d by a p e r i o d i c a r r a y o f d i s l o c a t i o n s w i t h spac ing dD, i s r e l a t e d t o d . The p resen t r e s u l t s suggest t h a t f o r t h e cases d iscussed here t h e s t r u c t u r a l w ie th , as determined by e l e c t r o n d i f f r a c t i o n , i s equal t o dD. For l a r g e ang le boundar ies i t i s more a p p r o p r i a t e t o t h i n k o f dD as

be ing t h e 0 - l a t t i c e p e r i o d i c i t y , i n s t e a d o f t h e d i s l o c a t i o n spacing. I t i s p o s s i b l e t h a t t h e presence o f seg rega t i on t o t h e g r a i n boundary w i l l

a f f e c t t h e r e l r o d p r o f i l e s . I n o r d e r t o use t h e d i f f r a c t i o n techn ique t o s tudy t h e d i s t r i b u t i o n o f s o l u t e i n t h e v i c i n i t y o f t h e boundary plane, i t w i l l be impor tan t t o understand i n a q u a n t i t a t i v e manner t h e f a c t o r s t h a t a f f e c t t h e i n t e n s i t y d i s t r i b u t i o n a long t h e r e l r o d .

V. General Conclusions A. D i f f r a c t i o n techn iques 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 d e t a i l e d

a tomic s t r u c t u r e o f l a r g e ang le g r a i n boundar ies. I n p a r t i c u l a r i t i s p o s s i b l e t o determine 1) t h e symmetry and, f rom t h i s , 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 s t r u c t u r e , 2) t h e p r o j e c t e d a tomic s t r u c t u r e and 3) t h e s t r u c t u r a l w i d t h o f t h e boundary.

B. The c=5(e=36.g0) and 1=13(0=22.6") [OOl] t w i s t boundar ies i n Au were shown t o be i n t h e u n t r a n s l a t e d ( co inc idence ) c o n f i g u r a t i o n , by a d e t e r m i n a t i o n o f t h e boundary symmetry.

C. The p r o j e c t e d s t r u c t u r e o f t h e Z=5 boundary c o n s i s t s o f groups o f atoms which have undergone l a r g e r o t a t i o n s about 0-elements i n t h e p lanes immedia te ly ad jacen t t o t h e boundary. The s t r u c t u r e i s made up o f d i scon t i guous patches of median f .c.c. s t r u c t u r e . A s i m i l a r a tomic c o n f i g u r a t i o n i s p resen t i n t h e ~ 1 3 boundary. It i s suggested t h a t l a r g e r o t a t i o n s about 0-elements t o produce median f.c.c. r eg ions occu r i n a l l l a r g e ang le [001] t w i s t boundar ies.

D. The a tomic s t r u c t u r e s o f t h e l a r g e ang le c=13 [OOl] t w i s t boundary i n Au and Ag a r e q u i t e s i m i l a r . T h i s conc lus ion i s i n disagreement w i t h computer m o d e l l i n g p r e d i c t i o n s t h a t t h e s t r u c t u r e s a r e q u i t e d i f f e r e n t . Comparison o f t h e c=5 s t r u c t u r e determined by d i f f r a c t i o n w i t h t h a t generated by computer m o d e l l i n g a1 so shows l a r g e disagreements.

E. The va lue ob ta ined f o r t h e s t r u c t u r a l w i d t h o f [ O O l ] t i l t boundar ies i n Au was shown t o be equal i n magnitude t o t h e d i s l o c a t i o n spac ing ( 0 - l a t t i c e p e r i o d i c i t y ) assoc ia ted w i t h t h e m i s o r i e n t a t i o n o f t h e boundary.

Acknowledgements T h i s resea rch was suppor ted by t h e Na t i ona l Science Foundat ion under g r a n t

DMR-79-16331 and i t s predecessors. The X-ray exper iments were c a r r i e d o u t a t t h e C o r n e l l High Energy Synchro t ron Source. The use o f t h e c e n t r a l f a c i l i t i e s o f t h e M a t e r i a l s Science Center i s g r a t e f u l l y acknowledged.

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