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GEOMETRY AND SYMMETRY OF THE (10,2) TWIN COINCIDENCE ORIENTATION IN CLOSE
PACKED HEXAGONAL STRUCTURE
S. Hagege, G. Nouet
To cite this version:
S. Hagege, G. Nouet. GEOMETRY AND SYMMETRY OF THE (10,2) TWIN COINCIDENCE
ORIENTATION IN CLOSE PACKED HEXAGONAL STRUCTURE. Journal de Physique Colloques,
1985, 46 (C4), pp.C4-237-C4-242. �10.1051/jphyscol:1985426�. �jpa-00224676�
GEOMETRY AND SYMMETRY OF THE (l0.2) TWIN COINCIDENCE ORIENTATION IN CLOSE PACKED HEXAGONAL STRUCTURE
S. Hagege and G. Nouet
Equipe Matériaux-Miorostrueture, Laboratoire de Cristallographie, Chimie et Physique des Solides^, ISMRa-Université, 14032 Caen Cedex, France
Résumé : Les notions de réseau dichromatique et de complexe dichromatique sont utilisées pour préciser la géométrie et la symétrie des joints de macle (10.2) dans la structure hexagonale compacte. Une approche géométrique en terme de coïncidence permet de préciser les caractéristiques des macles de déformation. La symétrie des différents complexes dichromatiques que l'on peut obtenir pour cette orientation de coïncidence est reliée à la structure hexagonale compacte.
Abstract : Geometry and symmetry of (10.2) twin boundaries in close packed hexagonal structure is analyzed through the notion of dichromatic pattern and dichromatic complex. The coincidence approach analyzes the geometric characteristics of deforma- tion twins. The symmetry of the different dichromatic complexes which correspond to the twin orientation is related to the close packed hexagonal structure.
INTRODUCTION
All the recent advances concerning the interfacial structure have been frequently limited on their applications to cubic.symmetry : simple cubic, bcc, fee, diamond ..
(1-4). It could be fruitful to look for lower symmetry structures in order to have a better understanding of their applicability and consequences. It is shown in this paper, by using the example of hexagonal symmetry, that new .insights can be given to the structure of deformation twin boundaries.
Deformation twins are very frequent in all close packed hexagonal metals and they are a simple case of coincidence boundaries if the hexagonal structure is described by a rational value of (c/a)2. The close packed hexagonal structure is described in the non symmorphic holoedral space group P 6,/mmc. The (10.2) twin coincidence orientation produces different dichromatic complexes related to one another by translations. Their symmetries indicate that this is not the classic mirror configu- ration having the highest symmetry, but another configuration, for which the centre of symmetry of each crystal is directly related by the rotation describing the (10.2) twin orientation.
The content of this work has been restricted to the study of the symmetry of the dichromatic patterns and complexes for the (10.2) twin coincidence orientation in cph structure. Its natural prolongation to the symmetry of bicrystals will be detai- led later on.
I - THE (10.2)TWIN IN THE CLOSE PACKED HEXAGONAL STRUCTURE
The establishement of ideal coincidence tables for hexagonal symmetry has shown that for any u/v ratio (u/v = (c/a)2) mechanical twins can be described as an ideal coincidence (5). In choosing a ratio u/v which is close enough to the experimental value (c/a)2of a metal, one can estimate precisely all the elements of the twin (6).
Twinning in close packed hexagonal (cph) metals is a necessary component of their deformation. One or more types of twinning are shown for every metal. This may be related to the c/a ratio by way of coincidence tables for y/v ratios (7-8). Descri- bing a particular metal by a u/v ratio attributes a Z value to every type of twin-
+ L.A. 251
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985426
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n i n g . The most f r e q u e n t t y p e (10.2) i s always r e l a t e d t o a low C v a l u e . See t a b l e I f o r r e s u l t s .
T a b l e I
-
(10.2) t w i n c o i n c i d e n c e i n d i c e s f o r v a r i o u s cph m e t a l s . M e t a lc i a
'lV
1.1. The geometry of t h e (10.2) t w i n c o i n c i d e n c e o r i e n t a t i o n
--
Be T i Zr Re Mg Co Zn Cd
1.568 1.587 1.595 1.615 1.623 1.623 1.856 1.886
5 / 2 1817 2118 813 311 2417 712 1815
1.581 1.604 1.620 1.633 1.732 1.852 1.871 1.897
A t w i n may be c h a r a c t e r i z e d by t h e o r i e n t a t i o n r e l a t i o n s h i p b r i n g i n g one c r y s t a l e l e m e n t i n t o c o i n c i d e n c e w i t h t h e o t h e r c r y s t a l e l e m e n t . F o r s i m p l e h e x a g o n a l symme- t r y 1 2 e q u i v a l e n t d e s c r i p t i o n s may be deduced. Each e q u i v a l e n t d e s c r i p t i o n i s d e - f i n e d by a r o t a t i o n a n g l e and a r o t a t i o n a x i s . For s i m p l e r e a s o n of c l a r i t y t h e r a t i o p/v= 3 w i l l be chosen f o r t h e p r e s e n t a t i o n of t h e r e s u l t s . Moreover t h i s r a t i o c o r r e s p o n d s t o t h e l o w e s t p o s s i b l e c o i n c i d e n c e i n d e x C = 2 f o r (10.2) t w i n o r i e n t a - t i o n .
1.1.1. The 1 2 e q u i v a l e n t d e s c r i p t i o n s
---
C (10.2) 11 1 3 15 1 7 2 15 1 3 11
I
T a b l e I1
-
The 12 e q u i v a l e n t d e s c r i p t i o n s a r e p l a c e d by i n c r e a s i n g v a l u e s of t h e r o t a t i o n a n g l e . The M i l l e r i n d i c e s of t h e r o t a t i o n a x i s a r e g i v e n w i t h t h e M i l l e r i n d i c e s of t h e p l a n e normal t o t h e r o t a t i o n a x i s .L
1.1.2. CSL and DSCL
----
C o n s i d e r i n g o n l y t h e B r a v a i s l a t t i c e o f t h e s t r u c t u r e ( s i m p l e h e x a g o n a l ) one c a n c a l c u l a t e t h e d i f f e r e n t u n i t c e l l s c o r r e s p o n d i n g t o t h e 0 - l a t t i c e , t h e c o i n c i d e n c e s i t e l a t t i c e (CSL) and t h e d i s p l a c e m e n t s h i f t c o m p l e t e l a t t i c e (DSCL) ( 9 ) .
i) t h e u n i t v e c t o r s of t h e 0 l a t t i c e a r e :
[-1 -112 1 / 2 1 , [ l 1 1 2 1 / 2 1 and [ O 1 1 2 01 i i ) t h e u n i t v e c t o r s of t h e CSL a r e :
[ O 0 11, [ 2 1 0 1 and [ O 1 01.
For a s i m p l e hexagonal t h e u n i t c e l l formed by t h e s e u n i t v e c t o r s shows a t e t r a g o n a l symmetry (P 4/mmm) b u t when i n t r o d u c i n g a second atom f o r t h e c l o s e packed h e x a g o n a l s t r u c t u r e t h i s u n i t c e l l does n o t c o n t a i n a l l t h e symmetry e l e m e n t s of t h e a t o m i c a r r a n g e m e n t . Then a d o u b l e c e l l w i l l b e c h o s e n t o r e p r e s e n t t h e CSL and
it
w i l lemphasize t h e r o l e p l a y e d by t h e t w i n p l a n e (10.2) and t h e s h e a r d i r e c t i o n C2111.
The u n i t v e c t o r s of t h e d o u b l e c e l l a r e :
[-2 -1
11,
[ 2 1 11 and [ O 1 O] ( ~ i g . 1 ) . i i i ) t h e u n i t v e c t o r s of t h e DSCL a r e :[-1 -112 0 1 , [ 0 0 1 / 2 1 and [O 1 01
For t h e same r e a s o n of s e t r y a s i n i i ) a d o u b l e c e l l w i l l b e c h o s e n : -112 1 / 2 1 , [ 1 112 1 / 2 1 and [ O 1 01 ( F i g . 1 )
1.1.3. D i c h r o m a t i c D a t t e r n
' 2" 0 8 0 I)
The d i c h r o m a t i c p a t t e r n ( d c p ) i s t h e p a t t e r n o b t a i n e d by v v e r l a p p i n g t h e B r a v a i s l a t t i c e s of c r y s t a l s 1 and 2 a f t e r t h e r o t a t i o n o p e r a t i o n . T h i s dcp i s n o t u n i q u e f o r t h e (10.2) twin c o i n c i d e n c e o r i e n t a t i o n . As p o i n t e d o u t by many a u t h o r s ( s e e ( 1 , 2 ) f o r example) a n i n f i n i t e number of d i f f e r e n t d c p ' s c a n b e c r e a t e d by t h e t r a n s l a t i o n of c r y s t a l 2. For a t r a n s l a t i o n t = 0 t h e p a t t e r n h a s t h e h i g h e s t symmetry mmm ( F i g . 2 a ) . F o r a l l t h e d i f f e r e n t p / v r a t i o s of t a b l e I t h e p a t t e r n s a r e d e s c r i b e d i n t h e f o l l o w i n g s p a c e g r o u p s :
I m l m ' m p l v = 1817 ; 2417 ; 1815 ; 813 B m ' m ' m p / v = 5 1 2 ; 2118 ; 712
C m l m ' m p / v = 3 ( F o r t h i s r a t i o t h e p a t t e r n i s e x c e p t i o n n a l l y t e t r a g o n a l : P & ' / m ' m ' )
.
1.2. Dichromatic complexes
-
*
0 0*
0
*
When i n t r o d u c i n g t h e s t r u c t u r e of cph m e t a l s , t h e r e a r e 2 atoms p e r u n i t c e l l and t h e 2 o v e r l a p p i n g c r y s t a l s form now t h e d i c h r o m a t i c complex ( d c c ) . The u s u a l way of p r e s e n t i n g t h i s d c c i s t o c h o o s e t h e m i r r o r o r i e n t a t i o n where t h e a t o m i c p o s i t i o n s of t h e two c r y s t a l s a r e i n m i r r o r p o s i t i o n r e l a t i v e t o t h e boundary p l a n e (10.2) ( F i g . 2b).
An i n f i n i t e number of d c c l s c a n be c r e a t e d by t r a n s l a t i n g c r y s t a l 2 by a v e c t o r t.
I f t i s a DSCL v e c t o r a n i d e n t i c a l dcc i s o b t a i n e d . I f t i s n o t a DSCL v e c t o r t h e dcc i s d i f f e r e n t and is c h a r a c t e r i z e d by t h e e q u i v a l e n t p o s i t i o n t ' of t i n s i d e t h e W i n e r S e i t z c e l l of t h e DSCL. F i u r e 3 i s o b t a i n e d f o r p / v = 3 by a r o t a t i o n of 90"
( c f o c k w i s e ) of c r y s t a l 2 around t i e r o t a t i o n a x i s [OIO] ( p o i n t i n g o u t of t h e p l a n e of t h e f i g u r e ) . F o r a r o t a t i o n a x i s l o c a t e d on a n a t o m i c p o s i t i o n ( 0 , 0 , 0 ) and t 1 = 0 t h e d c c i s d e s c r i b e d by t h e s p a c e group C m'2'm. The m i r r o r p l a n e m ' ( a n t i s y m m e t r y ) i s p a r a l l e l t o t h e (10.2) p l a n e and t h i s c o n f i g u r a t i o n i s t h e u s u a l way of p r e s e n - t i n g t h e dcc a s s o c i a t e d w i t h t h e (10.2) t w i n i n cph m e t a l s .
*
I h
0
( 1 ~ 2 ) 8
---
-
211 010-
I
F i g u r e 1 : ( l e f t ) R e p r e s e n t a t i o n of t h e d o u b l e c e l l s of t h e CSL and DSCL on s i m p l e hexagonal s t r u c t u r e f o r t h e (10.2) t w i n c o i n c i d e n c e o r i e n t a t i o n , P / v = 3.
F i g u r e 2 : ( a b o v e ) P r o j e c t i o n on t h e ( i 2 . 0 ) p l a n e of one d o u b l e c e l l of t h e CSL and 4 d o u b l e c e l l s of t h e DSCL.
a ) D i c h r o m a t i c p a t t e r n f o r s i m p l e hexagonal s t r u c t u r e .
b) D i c h r o m a t i c complex f o r c l o s e packed hexa- g o n a l s t r u c t u r e .
C r y s t a l 1 i s d e n o t e d by a s t a r and c r y s t a l 2 by a diamond. 0 c i r c l e d atoms a r e i n t h e p l a n e
010
of t h e f i g u r e ( z = 0 ) ; u n c i r c l e d atoms a r e o u t of t h e p l a n e of t h e f i g u r e ( z =*
1 1 2 ) . F i l l e d symbols a r e atoms i n c o i n c i d e n c e . Crys-100
t a l 1 i n d i c e s a r e r e p o r t e d o n t h e f i g u r e . 0I I
8
*
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Figure 3 : One dichromatic complex of the (10.2) twin coincidence orientation for the cph structure and p/v = 3 . The Wigner Seitz cell of the DSCL is centered in the middle of the CSL cell and is indexed by
X I , y', z' ranging between -112 to 112. t'
translation vectors are noted on the x' y'
z ' reference system. Same notations as in
fig,2.
1.2.1. The variation of the symmetry of the dcc with displacements of crystal 2.
--
Starting with the C m12m configuration (Fig. &a), translations parallel to x' will conserve the antisymmetry elements m' and P'and the ordinary mirror m. The corres- ponding dcc's are still described by the space group C m12'm even if atoms are no longer in coincidence.
For t' = (0, 112, 0) the antisymmetry mirror m' is destroyed and replaced by an antisymmetry mirror c' and the space group is then C c'2'm (the antisymmetry binary axis is translated to the level z = 114).
For some particular displacements t' new elements of symmetry can occur :
-
t 1 = (-112, -112, 112); the space group is now C 2'm'm (see Fig. 4b). This trans- lation can be described as if one atom of crystal 2 in a Wickoff position 2d of P 63/mmc (space group of the cph structure) is brought into coincidence with the other atom in the 2d Wickoff position of crystal 1 and creates a "stacking fault" like displacement.-
t' = (-116, 0 , 112). For this translation the dcc is described by the space group 'C m'm'm (Fig. 4c) which has a higher symmetry than the original dcc eventhough thisconfiguration does not have atom in coincidence.
Crystallographically equivalent variants of these dcc's arise if the order of the point symmetry of their space group is lower than that of the dcp (2). There are 2 variants for C m12'm, 2 variants for C 2'm'm and one for C m'm'm. The second variant is obtained by applying to the first variant the missing symmetry element m'.
It is worth noticing that for the (10.2) twin coincidence orientation ture this is not the (0, 0, 0) configuration which has the highest tentative explanation will be given in the following.
? '
in cph struc- symmetry. A
Figure 4 : Projection of some dichromatic complexes in one double cell of the CSL for the following displacements t' (Same notations as in Fig. 2) :
a. (0, 0, 0) ; space group C m'2'm b. (-112, -112, 112) ; space group C 2'm1m c - 6 0 , 2 ; space group C m'm'm
D i f f e r e n t d c c ' s have b e e n o b t a i n e d by d i s p l a c i n g t h e c r y s t a l 2 by v a r i o u s v e c t o r s t ' . Then t h e f i n a l o r i e n t a t i o n r e l a t i o n s h i p i s t h e p r o d u c t of a p u r e r o t a t i o n c e n t e r e d i n ( 0 , O ) by a t r a n s l a t i o n t l . A s i t i s w e l l known ( 1 0 ) t h i s o r i e n t a t i o n c a n be d e s c r i b e d d i r e c t l y by a p u r e r o t a t i o n , i f t ' i s a v e c t o r normal t o t h e r o t a t i o n a x i s . The c e n t r e o f r o t a t i o n i s d i s p l a c e d by t l . I f t 1 is n o t normal t o t h e r o t a t i o n a x i s , t h e c e n t r e w i l l be d i s p l a c e d by t h e p r o j e c t i o n of t 1 on t h e p l a n e normal t o t h e r o t a t i o n a x i s and t h e component p a r a l l e l t o t h e r o t a t i o n a x i s w i l l h a v e t o b e added t o t h e d i r e c t r o t a t i o n .
L e t u s c o n s i d e r t h e 2 c r y s t a l s b e f o r e r o t a t i o n when a l l t h e atoms a r e i n c o i n c i d e n c e ( F i g . 5 ) .
F i g u r e 5 : P r o j e c t i o n o f t h e cph s t r u c t u r e I i n one d o u b l e c e l l o f t h e CSL. Atom A i n ( 0 , 0 , 0 ) i s o n t h e Wickoff p o s i t i o n 2 d ( 2 ) . Atom B i n (5 1 6 , 1 1 6 , 0 ) i s on t h e Wickoff p o s i - t i o n 2 d ( l ) . Black d o t s a r e p o s s i b l e p o s i t i o n of t h e c e n t r e of symmetry of t h e s t r u c t u r e . X a r e p o s s i b l e c e n t r e s o f d i r e c t r o t a t i o n .
- -8 Same n o t a t i o n s a s i n F i g . 2.
F i g u r e & a i s o b t a i n e d by a d i r e c t r o t a t i o n c e n t e r e d o n A ( 0 , 0 ) i.e. a n a t o m i c p o s i - t i o n . E q u i v a l e n t d c c ' s w i l l be o b t a i n e d f o r a d i r e c t r o t a t i o n c e n t e r e d on any a t o m i c p o s i t i o n s and t h e i r e q u i v a l e n t i n t h e Wigner S e i t z c e l l ( 1 / 2 , - 1 1 2 , 1 1 2 ) ( - 1 / 6 , 1 / 6 , 0 ) ( 1 1 3 , -113, 1 1 2 ) . I n t h e cph s t r u c t u r e a l l a t o m i c p o s i t i o n s a r e c r y s t a l l o g r a p h i c a l l y e q u i v a l e n t . I t i s t h e n o b v i o u s t h a t t h e c o r r e s p o n d i n g d c c ' s a r e e q u i v a l e n t . They a r e d e s c r i b e d by t h e s p a c e g r o u p C m'2'm.
F i g u r e 4b c a n be o b t a i n e d by a d i r e c t r o t a t i o n c e n t e r e d o n Xl ( 1 1 3 , 1 / 2 1 . T h i s o p e r a t i o n b r i n g s d i r e c t l y t h e atom B ( 5 1 6 , 1 1 6 , 0 ) i n t o c o i n c i d e n c e w i t h a n o t h e r one A ( 0 , 0 , 0 ) . A l l e q u i v a l e n t d i r e c t r o t a t i o n s b r i n g i n g o n e atom i n t o c o i n c i d e n c e w i t h a n o t h e r one i n t h e s t r u c t u r e w i l l p r o d u c e e q u i v a l e n t d c c ' s b e l o n g i n g t o t h e s p a c e g r o u p C"2'm1m.
F i g u r e 4c c a n be o b t a i n e d by a d i r e c t r o t a t i o n c e n t e r e d o n X (-1112, 1 / 1 2 ) . For t h e cph s t r u c t u r e one can d e f i n e a c e n t r e of symmetry which can 8 e p l o t t e d on F i g . 5 on one o f t h e f o l l o w i n g p o i n t s ( b l a c k d o t s ) ( 5 1 1 2 , 1 / 1 2 , 0 )
,
(-1112, -5112, 0 ) ( 5 1 1 2 , 1 / 1 2 , 1 1 2 ) o r (-112, -5112, 1 / 2 1 . The d i r e c t r o t a t i o n c e n t e r e d on ( - 1 / 1 2 , 1 / 1 2 ) b r i n g s two of t h e s e p o s i t i o n s i n t o c o i n c i d e n c e and d e f i n e s a c o i n c i d e n c e" s i t e " l a t t i c e w i t h o u t atom i n c o i n c i d e n c e and l e a d s t o a h i g h e r symmetry s p a c e g r o u p C m ' m ' m .
11 - GEOMETRY AND STRUCTUE OF THE (10.2) TWIN I N cph METALS
I n t h e p r e c e e d i n g p a r t of t h i s p a p e r t h e r a t i o p / v = 3 has been c h o s e n t o c l a r i f y t h e d i f f e r e n t a s p e c t s o f t h e geometry of t h e ( 1 0 . 2 ) t w i n c o i n c i d e n c e o r i e n t a t i o n . (10.2) d e f o r m a t i o n t w i n s o c c u r s i n a l l cph m e t a l s and t h e a m p l i t u d e of t h e s h e a r i s d i r e c t l y f u n c t i o n of t h e c / a r a t i o and more e s p e c i a l l y f u n c t i o n o f t h e d e v i a t i o n from t h e i d e a l r a t i o c l a = J 3 f o r which t h e s h e a r i s z e r o . I n o r d e r t o p r e s e n t t h e v a l i d i t y of t h e s e r e s u l t s t h e c a s e of z i n c ( c / a = 1.856) w i l l b e d e s c r i b e d by a r a t i o p / v = 712 ( s e e t a b l e I ) . Moreover t h e a u t h o r s w i l l t a k e t h i s o p p o r t u n i t y t o c o r r e c t some e r r o n e o u s a f f i r m a t i o n s and f i g u r e s o f o n e o f t h e i r p r e v i o u s p a p e r ( 6 ) .
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JOURNAL DE PHYSIQUE
For u/v = 712 the coincidence twin orientation is described by a rotation of 85"59 around the [OlO 1 axis and a coincidence index X = 13.
The vectors of the CSL are : [-2 -1 1
1,
[ 1 4 7 6 l a n d C O 1 0 1 of the DSCL are : 1/13 [-2 -1 11, 1/13 [ 14 7 6 1 and [0 1 01 (note the different values for the DSCL vectors compared to (6)).Among all the dccls one can obtain for this coincidence orientation three particular configurations can be described (Fig. 6) :
a) mirror configuration, space group B m12'm. Fig. 6a is the corrected version of Fig. 4 in (6).
b) "stacking fault1' like configuration, space group B 2'm1m (Fig. 6b).
c) highest symmetry with centre of symmetry in coincidence, space group B m l m l m (Fig. 6c).
Figure 6 : 2/13 .of the double cell of the CS1 foru / v = 712 marking the symmetry of 3 different dichromatic patterns for the (10.2) twin orientation. (Same notations as in Fig. 2 )
ACKNOWLEDGEMENTS
This work has been performed under the auspices of the association ISMRa-CENISCK. We are indebted to P. Delavignette for his continuous interest in this study and for drawing our attention to the C 2'rn'm configuration of the (10.2) twin coincidence orientation.
REFERENCES
1) R.C. POND and W. BOLLMANN, Phil. Trans. R. Soc. A, 292, 449 (1979).
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