Structure of grain boundaries
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an a r r a y of d i s l o c a t i o n s than in t e r m s of coincidence sites. 2) At c e r t a i n a n g u l a r r e l a t i o n s h i p s between the c r y s t a l s , s i t e s of the two c r y s t a l s coincide at the b o u n d a r y . An example is shown in Fig. l(a) which is a 28.1 deg b o u n d a r y about a axis. It is a p r o p e r t y of r e l a tionships such as this that, if one of the two l a t t i c e s is extended to o v e r l a p the other, t h e r e is a t h r e e dim e n s i o n a l a r r a y (or lattice) of c o i n c i d e n c e s . However, the only atoms that a r e c o m m o n to the two l a t t i c e s a r e those at the b o u n d a r y itself. A s e r i e s of c o i n c i dence r e l a t i o n s h i p s of this kind can be developed for r o t a t i o n s about (100>, , and (110> axes. 3) The s l i g h t e s t d e p a r t u r e f r o m the g e o m e t r i c a l l y exact r e l a t i o n s h i p outlined above should c o m p l e t e l y d e s t r o y the coincidence lattices, but it i s p o s s i b l e to r e t a i n b o u n d a r y coincidence at the expense of the p r e cise p e r i o d i c i t y of coincidence s i t e s that c h a r a c t e r izes the " i d e a l c o i n c i d e n c e " r e l a t i o n s h i p defined in 2) above. Thus any angle b e t w e e n 28 and 37 deg about [100] can be r e p r e s e n t e d by the a p p r o p r i a t e r a tio of the two e l e m e n t s of s t r u c t u r e shown in Fig. l(b). An e x a m p l e is shown in Fig. l(c). The m a i n t e n a n c e of c o i n c i d e n c e in the plane of the b o u n d a r y r e q u i r e s some d i s t o r t i o n of the c r y s t a l s ; this i n c r e a s e s with i n c r e a s i n g d e p a r t u r e from an "ideal coincidence" relationship. It can be shown that any b o u n d a r y can be r e p r e s e n t e d as a s p e c i a l case of the t r a n s i t i o n b e t w e e n " i d e a l c o i n c i d e n c e " a n g l e s . The r e p r e s e n t a t i o n of the bounda r y in t e r m s of coincidence s i t e s is not dependent on the b o u n d a r y b e i n g s y m m e t r i c a l l y d i s p o s e d between the two l a t t i c e s . The " b o u n d a r y c o i n c i d e n c e " m o d e l can a l s o be looked upon as having the g e o m e t r i c a l c h a r a c t e r i s t i c s of an a r r a y of d i s l o c a t i o n s with two d i s l o c a t i o n s for each c o i n c i d e n c e s i t e , Fig. l(d). However, it should be pointed out that for l a r g e a n g l e s , the d i s l o c a t i o n s a r e so close t o g e t h e r that t h e i r c o r e s overlap. It is t h e r e fore not v a l i d to p r e d i c t any p r o p e r t i e s of dislocation a r r a y s that a r e r e l a t e d to the c o r e s of individual d i s locations. On the other hand, the e l a s t i c s t r a i n field can be c a l c u l a t e d on the s a m e b a s i s as for the s m a l l angle b o u n d a r y . A t h i r d , and equally valid, d e s c r i p t i o n of the s t r u c t u r e u n d e r d i s c u s s i o n i s in t e r m s of ledges. T
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