HAL Id: jpa-00224675
https://hal.archives-ouvertes.fr/jpa-00224675
Submitted on 1 Jan 1985
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
COINCIDENCE ORIENTATIONS OF GRAINS IN HEXAGONAL MATERIALS
H. Grimmer, D. Warrington
To cite this version:
H. Grimmer, D. Warrington. COINCIDENCE ORIENTATIONS OF GRAINS IN HEXAG- ONAL MATERIALS. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-231-C4-236.
�10.1051/jphyscol:1985425�. �jpa-00224675�
Colloque C4, suppl6ment au n04, T o m e 46, avril 1985 page C4-231
COINCIDENCE ORIENTATIONS OF GRAINS IN HEXAGONAL MATERIALS
H. Grimmer and D.H. warrington+
Swiss Federal I n s t i t u t e for Reactor Research, CH-5303 WGrenZingen,
$witzerZand
Department of Metallurgy, S h e f f i e l d University, U . K .
Abstract - For arbitrary rotations expressed in hexagonal coordinates,the connection between an axis-angle quadruple and the rotation matrix is given.
The relation between hexagonally equivalent quadruples is stated and a unique reduced quadruple is defined. Rotations generating coincidence site lattices
(CSLs) correspond to quadruples of coprime integers if a2/c2 is rational. The multiplicity C of the CSL is not always equal to the least common denominator of the elements in the rotation matrix. The case a = c is considered in detail.
INTRODUCTION
The coincidence site lattice model of grain boundaries states that energetically favourable boundaries between two grains of the same phase are possible if a large portion 1/C of lattice vectors of one grain are simultaneously lattice vectors of the other grain. The 3-dimensional lattice formed by the common vectors of the two congruent lattices is called the coincidence site lattice (CSL), C is called its multiplicity, a rotation connecting two lattices with a common CSL is called a coin-
cidence rotation. For hexagonal lattices, such rotations have been investigated by several groups of authors [l- 71.
THE ROTATION MATRIX IN HEXAGONAL LATTICE COORDINATES
In a hexagonal coordinate system with lell = le21 = a, le31 =.c, &(el, e2) = 120°, an arbitrary rotation with angle 0 and axis [pl, p2, p31 is glven by the matrix
(
3~' + 2AD - 'D + r ( B ~ - c2) rB (2C - B) - 4AD 2(BD+A(2C- B)) R = 7 rC (2B - C) + 4AD ~ A ~ - ~ A D - D ~ - ~ ( B ~ - c ~ ) 2(CD-A(2B-C))
r(D(2B- C) - 3AC) r (D ( 2 ~ - B) + 3AB) ~ A ~ + D ~ - ~ ( B ~ - B c + c ~ )
where r = a2 / c2 (2)
0 0 0 0
and (A, B, C, D) = + (cos -, p sin -, p2 sin - r p3 sin 7
2 1 2 2
if the axis [pl, p2, p31 is normed to have length &c, i.e. if
It follows that the quadruple (A, B, C, D) satisfies the normalization condition
The normalized quadruples (A, B, C, D) describe rotations in hexagonal coordinates in the same way as quaternions describe rotations in orthonormal coordinates [8l.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985425
C4-232 JOURNAL DE PHYSIQUE
EQUIVALENT ROTATIONS, CHOICE OF A UNIQUE REDUCED ROTATION I N EACH EQUIVALENCE CLASS The r e l a t i v e o r i e n t a t i o n between two n e i g h b o u r i n g g r a i n s o f t h e same p h a s e w i t h p o i n t g r o u p 622 o r 6/mmm c a n b e d e s c r i b e d i n d i f f e r e n t ways: c a r r y o u t o n e o f t h e 1 2 symmetry r o t a t i o n s o f o n e g r a i n , t h e n a r o t a t i o n t h a t o r i e n t s i t p a r a l l e l t o t h e o t h e r g r a i n , t h e n a symmetry r o t a t i o n i n i t s new o r i e n t a t i o n . Because a l s o t h e r o l e s o f t h e t w o g r a i n s c a n b e i n t e r c h a n g e d , up t o 2 . 1 2 2 = 288 d i f f e r e n t r o t a t i o n s a r e o b t a i n e d . They w e r e c a l l e d ( h e x a g o n a l l y ) e q u i v a l e n t by Grimmer [91, who showed t h a t t h e number n o f d i f f e r e n t r o t a t i o n s i n any e q u i v a l e n c e c l a s s h a s t h e form n = 1 2 w , where w is a n i n t e g e r d i v i d i n g 24. An example w i t h w = l i s t h e c l a s s c o n s i s t i n g o f t h e 1 2 h e x a g o n a l symmetry r o t a t i o n s . T h e r e a r e 2n d i f f e r e n t e q u i v a l e n t q u a d r u p l e s b e c a u s e o f 2 i n ( 3 ) .
TABLE 1: 1 2 e q u i v a l e n t q u a d r u p l e s r e p r e s e n t i n g t h e d i f f e r e n t r o t a t i o n a n g l e s t h a t o c c u r i n a n e q u i v a l e n c e c l a s s . They a r e o b t a i n e d by l e t t i n g o p e r a t o r s I , J , K , L a c t o n t h e q u a d r u p l e q = (A, B, C, D ) .
T a b l e 1 g i v e s 1 2 q u a d r u p l e s r e p r e s e n t i n g t h e d i f f e r e n t r o t a t i o n a n g l e s t h a t o c c u r i n a n e q u i v a l e n c e c l a s s . The 576 e q u i v a l e n t q u a d r u p l e s a r e o b t a i n e d from t h o s e i n t h e t a b l e by a r b i t r a r y c o m b i n a t i o n s o f t h e f o l l o w i n g 4 o p e r a t i o n s
9 I S Jq
~ q IKq JKq Lq ILq J L ~ KLq IKLq JKLq
a ) s i g n change o f t h e 1st component b ) s i g n change o f t h e 4 t h component C ) i n t e r c h a n g i n g t h e 2nd and 3 r d component
d ) r e p l a c i n g t h e 2nd a n d 3 r d component B, C by B - C , B. Applying t h i s r e p e a t e d l y o n e o b t a i n s B, C -+ B - C , B -t - C , B - C + - B , - C -+ C - B , - B -+ C, C - B + B , C.
(A B C D )
1/2 (A+ D 2B 2C 3A-D )
1/2- (D-A 2B 2C 3A+D )
1/43 (D B - 2C 2B- C 3A )
1 / ( 2 & ) (3A+D 2 ( B - 2C) 2 ( 2 B - C ) 3 ( D - A ) ) 1 / ( 2 & ) (3A-D 2 ( B - 2C) 2 ( 2 B - C ) 3 ( A + D ) ) 1 / ( 2 & ) ( ( B - C ) r 2 ( A + D ) 4A ( B + C ) r )
1 / ( 2 & ) (Br 2 ( A + D ) 4A (B - 2C) r )
1 / ( 2 & ) (Cr 2 ( A + D ) 4A (2B- C ) r )
2 ( ( B + C ) r 2 ( D - 3A) 4D 3 ( B - C ) r )
1 / ( 2 & ) ( ( 2 B - C) r 2(D- 3A) 4D 3Cr )
1 / ( 2 6 ) ( ( B - 2C) r 2 (D - 3A) 4D 3Br 1
Q u a d r u p l e s c o n n e c t e d by t h e s e o p e r a t i o n s c o r r e s p o n d t o r o t a t i o n s w i t h t h e same a n g l e and w i t h a x e s r e l a t e d by h e x a g o n a l symmetry o p e r a t i o n s : a ) i n v e r s i o n , b / c ) r e f l e c - t i o n s i n p l a n e s p e r p e n d i c u l a r / p a r a l l e l t o t h e 6 - f o l d a x i s , d ) 60° r o t a t i o n .
A u n i q u e r e d u c e d q u a d r u p l e i n e a c h e q u i v a l e n c e c l a s s i s d e t e r m i n e d by t h e c o n d i t i o n s
( 6 ) c h o o s e s among e q u i v a l e n t r o t a t i o n s o n e w i t h a x i s i n a s t a n d a r d s t e r e o g r a p h i c t r i a n g l e , ( 7 ) o n e w i t h minimum r o t a t i o n a n g l e . I f t h e r e a r e s e v e r a l s u c h r o t a t i o n s
( 8 a - c ) w i l l make a u n i q u e c h o i c e .
A r o t a t i o n g e n e r a t e s a CSL i f and o n l y i f i t s m a t r i x R e x p r e s s e d i n l a t t i c e c o o r d i - n a t e s i s r a t i o n a l , i . e . h a s o n l y r a t i o n a l m a t r i x e l e m e n t s [ l o ] . From ( 1 ) and ( 6 ) f o l l o w s a ) i f r is i r r a t i o n a l t h a t R c a n b e r a t i o n a l o n l y i f e i t h e r A = D = O o r B = C = 0 and b ) f o r any v a l u e o f r t h a t
I f t h e R i a r e r a t i o n a l and A # 0 , t h e l e f t hand s i d e s o f ( 9 ) w i l l b e r a t i o n a l be- c a u s e e i t i e r r is r a t i o n a l o r B = C = 0. Then t h e r e e x i s t s a r a t i o n a l number k and f o u r c o p r i m e i n t e g e r s m, U , V, W s u c h t h a t n2 = km, AB = kU, AC = kV, AD = kW.
"Coprime" means t h a t t h e g r e a t e s t common d i v i s o r o f t h e i n t e g e r s e q u a l s 1, i . e .
gcd ( m , U , V , W ) = 1 . (10)
Using ( 5 ) and d e f i n i n g S by
o n e o b t a i n s k = m/S and
I t i s e a s y t o show t h a t ( 1 0 ) and ( 1 2 ) remain t r u e i f A = 0 .
I f r i s r a t i o n a l , t h e r e e x i s t p o s i t i v e i n t e g e r s p, v s a t i s f y i n g and
P u t t i n g 2
F = 31-1s = w(3m2+w2) + v ( u 2 - W + V 1, - (15)
and u s i n g ( 1 2 ) o n e c a n w r i t e (1) a s
( 1 5 ) and (16) show t h a t t h e n e c e s s a r y c o n d i t i o n ( 1 0 ) i s a l s o s u f f i c i e n t ( f o r r a - t i o n a l r ) t o g u a r a n t e e t h a t R is r a t i o n a l and t h e r e f o r e a c o i n c i d e n c e r o t a t i o n . I t s a x i s i s [ u , V, W ] and i t s a n g l e i s g i v e n by c o s ( 8 / 2 ) = A = m/&. I f , i n a d d i t i o n , m=W=O o r U = V = 0 , R becomes i n d e p e n d e n t o f r . These r o t a t i o n s a r e c o i n c i d e n c e ro- t a t i o n s f o r e v e r y ( r a t i o n a l o r i r r a t i o n a l ) v a l u e o f r . I f m= W = 0 , t h e y a r e 180°
r o t a t i o n s a b o u t l a t t i c e v e c t o r s p e r p e n d i c u l a r t o t h e 6 - f o l d a x i s , i f U = V = O , t h e y a r e r o t a t i o n s by c e r t a i n a n g l e s a b o u t t h e 6 - f o l d a x i s . N o t i c e t h a t t h e n o r m a l i z a t i o n c o n d i t i o n ( 5 ) h a s been r e p l a c e d by ( 1 0 ) f o r c o i n c i d e n c e r o t a t i o n s .
THE MULTIPLICITY C OF THE CSL GENERATED BY A COINCIDENCE ROTATION R
Denote by C ' and C " t h e l e a s t common denominator o f t h e m a t r i x e l e m e n t s o f R and R -1 ,
r e s p e c t i v e l y . I t was shown i n [ l o , 111 t h a t C i s t h e l e a s t common m u l t i p l e o f C ' and Z " ,
c = lcm (11, c " ) . (17)
C = C * = C" f o r c u b i c l a t t i c e s . C o n t r a r y t o a w i d e l y h e l d view [l, 5 , 6 1 , t h i s r e - mains n o t a l w a y s t r u e f o r h e x a g o n a l l a t t i c e s .
C4-234 JOURNAL DE PHYSIQUE
As an example where I: # C' consider for c/a = 7 the rotation R given by +(-3,42, 14, 1) and R-1 given by 2 (3 , 42, 14, 1) . The corresponding matrices are
R generates a CSL with basis 1 1 = ( ) , 7 s2 = ( ) , e3 = ( ) .
Its multiplicity is C = 98 = lcm (C', C") # C' = 14.
An example where all 3 numbers C', C" and C are different is given by (77, 286, 130, 7) with c/a = 13/7. C = 7 C' = 13 C" in this case.
EQUIVALENCE CLASSES OF COINCIDENCE ROTATIONS: GENERAL RESULTS
The equations ( 6 - 8) become somewhat simpler for coincidence rotations:
For any given value of r , at least one of the two eqs. (20) can be dropped, too. It is now easy to determine complete lists of all equivalence classes with given values for p and for the maximum multiplicity of interest C,. This will be illustrated by considering the case c = a.
EQUIVALENCE CLASSES OF COINCIDENCE ROTATIONS: THE CASE c = a
In this case, which was discussed first in [41, FI = v = 1 and G = C' = C". To find all equivalence classes with X < Em determine the quadruples of coprime integers m, U, V, W satisfying (18- 20a) for r = l as well as one of the following four addi- tional conditions, which help to list the classes quickly (alb means a divides b) : 1) FIG,
2 ) F i 3 C m 3 1 ~ ~ 3 1 ~
3) F S 4Cm 41F , 210 , 2 1 ~ , m a n d W o d d
4) F S 1 2 C m , 121F , 2(u , 21V , modd, W odd multiple of 3.
The multiplicity C is given by C = F/12 if 12 1 F, 2 ) ~ , 2 ) ~ . Otherwise Z = F/4 if 4 ) ~ , 2)u, 2 1 ~ ; C = F/3 if 3 1 ~ . C = F in the remaining cases.
The result for Em = 12 is given in Table 2, where n = 1 2 w is the total number of rotations in the equivalence class and N(X) = 12W(X) the total number of rotations generating a CSL with multiplicity C.
hexagonal lattice with c = a.
W(1) = 1; W(C) = 0 if 31 C. In the remaining cases, consider the decomposition of C into a product of primes. If it contains k different primes,
then
The reduced rotation in the class
Examples :
m U V W
1 0 0 0
1 2 1 0
1 1 0 0
2 2 1 0
3 0 0 1
2 3 0 0
3 2 0 1
3 2 1 0
3 4 2 1
3 4 0 1
Properties of the class
The number n = 12w of rotations in a class and the number n l ~ o of 180° rotations are given in Table 3 for different forms of the reduced quadruple. The last column gives the symmetry of coincidence site lattice and DSC lattice [11.
0 I01 F
0 3
90 6
60 4
53.130 15 21.787 2 8 81.787 21 46.567 32 36.870 30 69.513 40 76.863 44
C
1 2 4 5 7 7 8 10 10 11
TABLE 3: Some properties determined by the form of the reduced quadruple.
monoclinic
m 2V V
m 2m V
w 1 3 6 6 6 12 12 12
The classes with w = 1 or 2 are those common to all values of r.
w(C) 1 3 6 6
' I 8 12
} 18
12
C4-236 J O U R N A L DE PHYSIQUE
CONNECTION WITH RESULTS OF OTHER AUTHORS
-
(10) and (13- 16) are equivalent to results given in [ 5 ] and [61. The transition is made by putting n = g c d (U, V, W) , u=U/n, v=V/n, w=W/n. The main difference is that in deriving (10) and (13-16) no use was made of the wrong assumption that C = 1' in all cases.
ACKNOWLEDGEMENTS
D.H. Warrington is grateful for support of NATO grant 1650.
REFERENCES
WARRINGTON, D.H., J. Physique 36 (1975) C4- 87.
FORTES, M.A. and SMITH, D.A., Scr. Metall. 10 (1976) 575.
BONNET, R., COUSINEAUl E. and WARRINGTON, D.H., Acta Cryst. A 37 (1981) 184.
HAGEGE S. , NOUET, G. and DELAVIGNETTE, P., phys. stat. sol. (a) 61 (1980) 97.
BLERIS, G.L., NOUET, G., HAGEGE, S. and DELAVIGNETTE, P., Acta Cryst. A s (1982) 550.
DELAVIGNETTE, P., J. Physique 43 (1982) C6- 1.
GRIMMER, H. and WARRINGTON, D.H., Z. Krist. 162 (1983) 88.
GRIMMER, H., Acta Cryst. A 2 (1974) 685.
GRIMMER, H., Acta Cryst. A 2 (1980) 382.
GRIMMER, H., Acta Cryst. A 3 (1976) 783.
FORTES, M.A., Acta Cryst. A 2 (1983) 351.
