An introduction to computations in crystallographic textures A.Morawiec Polish Academy of Sciences Institute of Metallurgy and Materials Science Motivation Comprehensive understanding of description of orientations is crucial for research on polycrystalline materials. Examples: • Orientation relationships • Crystal deformation mechanisms • Some phase transformation mechanisms http://www.museumwales.ac.uk • Orientation mapping Wikipedia.com 2 =20 µm; Euler3 + 3/ 10/ 20; Step=1 µm; Grid100x100 Outline • Rotations and rotation parameterizations • Crystal orientation and crystal symmetry • Statistics in the orientation space • Standard (mis)orientation distributions • Example of texture application: effective elastic properties of polycrystals 3 Suggested reading • H.J. Bunge, Texture Analysis in Materials Science, Butterworths, London, 1982. • U. F. Kocks, C. N. Tome, H.R. Wenk, Texture and Anisotropy, Preferred Orientations in Polycrystals and their Effect on Materials Properties, Cambridge University Press, Cambridge, 1998. • A. Morawiec, Orientations and Rotations, Computations in Crystallographic Textures, Springer Verlag, Berlin, 2004. 4 Rotations and rotation parameterizations A.Morawiec Polish Academy of Sciences Institute of Metallurgy and Materials Science 5 Outline • Basics: orientations vs. rotations • Numerical representation of rotations and orientations • Composition of rotations • Parameterizations of rotations and orientations Wikipedia.com 6 Basics Basics Rotation about a point – a displacement in which the location of the point is not changed. Rotation about a line – a displacement in which points of a line retain their locations. Euler theorem: Rotation about a point is equivalent to a rotation about a line Wikipedia.com 8 Basics An orientation – the equivalence class of all displacements which differ by a translation. With a universal reference orientation, an orientation of an object is determined by the rotation from the reference orientation to that orientation. object's orientations rotations about an axis one-to-one correspondence Orientation – state Rotation – process (displacement) 9 Basics ‘Mirror’ transformation with a fixed point = improper rotation Improper rotations change handedness Effects of proper rotations i.c Mirror images Effects of proper and improper rotations i.c 10 Basics Inversion Inversion = half-turn about a line followed by reflection with respect to a plane perpendicular to the line. improper rotation = proper rotation composed with inversion Wikipedia.com 11 Numerical representation of rotations and orientations To refresh memory … Matrix – an array of m n numbers Aij i 1,2,..., m j 1,2,..., n An m n matrix A is: square matrix if m = n zero if Aij = 0 transpose of B=AT if Aij = Bji A square matrix A is: unit if Aij = Iij = dij anti-symmetric if Aij = -Aji symmetric if Aij = Aji A11 A A 21 ... Am1 1 d ij 0 A12 A22 ... Am 2 A1n A2 n ... ... ... Amn ... ... if i j if i j 1 if ijk even permutation of (123) e ijk 1 if ijk odd permutation of (123) 0 in other cases Matrix algebra n Matrix product: C = AB Cij Aik Bkj Ai1B1 j Ai 2 B2 j ... Ain Bnj Aik Bkj k 1 Trace of square matrix Aij: Tr(A)= Aii = A11 + A22 + … + Ann Det of 3 3 matrix Aij: det(A)= eijk Ai1 Aj2 Ak3 Square matrix A is invertable if det(A) is non-zero; AA-1=I 13 Matrix representation of orientations p dim object immersed in N dim Euclidean space Orientation is determined by p linearly independent vectors N=3, p=2 a a1 a3 a2 b b1 b2 With orthonormal bases b3 a1 a2 a3 O b1 b2 b3 a a a b 1 0 OOT 0 1 b a b b OOT I p An orientation of a p dim object immersed in N dim Euclidean space can be represented numerically by a p x N matrix O satisfying OOT=I. 14 Matrix representation of orientations N=p = 3 OOT I 3 O – an orthogonal matrix det O2 1 OT O 1 Arbitrary rotations (including improper rotations) N=3, p=3 OOT I 3 det O 1 O – a special orthogonal matrix Proper rotations 15 Matrix representation of orientations N=p = 3 Example (special) orthogonal matrix 2 / 3 1/ 3 2 / 3 O 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 N=3, p=3 2 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 1 0 0 OOT 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 0 1 0 I 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 2 / 3 0 0 1 det O 1 16 Matrix representation of orientations N=p = 3 OOT I 3 O – an orthogonal matrix det O2 1 OT O 1 Arbitrary rotations (including improper rotations) N=3, p=3 OOT I 3 det O 1 O – a special orthogonal matrix Proper rotations Groups of orthogonal matrices – O(3) Wikipedia.com special orthogonal matrices – SO(3) 17 Composition of rotations Groups – all rotations O(3) one-to-one correspondence to SO(3) Wikipedia.com – proper rotations Composition of rotations R' O' OT O R' ' O' ' O'T O' O' ' R O' ' OT O, O’ and O’’ – orthogonal matrices R O' ' OT O' ' IOT O' ' (O'T O' )OT (O' ' O'T )(O' OT ) R' ' R' R R' ' R' CompositionWikipedia.com of rotations corresponds to multiplication of representing them orthogonal matrices. 19 Composition of rotations 2 / 3 1/ 3 2 / 3 R' 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 0 1 0 R' ' 1 0 0 0 0 1 0 1 0 2 / 3 1 / 3 2 / 3 2 / 3 2 / 3 1 / 3 R' ' R' 1 0 0 2 / 3 2 / 3 1 / 3 2 / 3 1 / 3 2 / 3 R 0 0 1 1 / 3 2 / 3 2 / 3 1 / 3 2 / 3 2 / 3 R R' ' R' CompositionWikipedia.com of rotations corresponds to multiplication of representing them orthogonal matrices. 20 Composition of rotations − algebra of quaternions e - basis of a 4 dimensional vector space, 0,1,2,3 i, j, k 1,2,3 x x e x0e0 x1e1 x2e2 x3e3 Standard multiplication plus the quaternion multiplication rule { x x e e e0 e0e e ei e j e ijk ek d ij e0 y y e xy x0 y0 xi yi e0 x0 yk xk y0 e ijk xi y j ek xy yx Wikipedia.com 21 Unit quaternions and rotations q - components of a unit quaternion i, j, k 1,2,3 0,1,2,3 q q q02 q12 q22 q32 1 Aij q0 qk qk d ij 2qi q j 2e ijk qk q0 2 A is a special orthogonal matrix q [ q0 , q1 , q2 , q3 ] [ 1 / 2, 1 / 2, 1 / 2, 1 / 2 ] 0 0 1 A 1 0 0 0 1 0 22 Unit quaternions and rotations q, q’ – unit quaternions O, O’ – special orthogonal matrices q O q ' O' qq' OO' There is a two-to-one corespondence between unit quaternions and special orthogonal matrices. Wikipedia.com There is a two-to-one corespondence between unit quaternions and proper rotations. 23 Unit quaternions and rotations Unit quaternion sphere in 4D q q02 q12 q22 q32 1 q Quaternions q and –q represent the same orientations Oij q q0 qk qk d ij 2qi q j 2e ijk qk q0 Oij q 2 24 Summary Euler theorem. (Proper) rotations can be represented (special) orthogonal matrices. Composition of rotations is represented by the product of orthogonal matrices. There is two-to-one correspondence between unit quaternions and special orthogonal matrices. Composition of rotations is represented by the product of unit quaternions. Wikipedia.com 25 Transformation of vector components x ai a j d ij and x x j a j x'k a 'k a'i a' j d ij a'i x'i x j a j a'i Rij x j a'i a j Rij x' Rx Wikipedia.com 26 Parameterizations • Rodrigues parameters • Axis and angle • Rotation vector • Euler angles • Miller indices Independent orientation parameters Special orthogonal matrix – 9 parameters Unit quaternion – 4 parameters 1 2 3 4 1 2 3 4 5 6 7 8 9 Number of independent parameters - 3 Is it possible to have a „nice” global parameterization (one-to-one, continuous, with continuous inverse), which would map rotations into the 3 dimensional Euclidean space? No! Wikipedia.com 28 Cayley transformation O - special orthogonal matrix such that ( I O) non-singular R ( I O) 1 ( I O) O ( I R)( I R) 1 R - an antisymmetric 3x3 matrix 3 independent parameters ( I R) 0 R r3 r2 non-singular r3 0 r1 r2 r1 0 29 Cayley transformation, example 2 / 3 1/ 3 2 / 3 O 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 1 / 3 1 / 3 0 0 R ( I O) 1 ( I O) 1 / 3 0 1 / 3 r3 1 / 3 1 / 3 0 r2 r3 0 r1 r2 r1 0 r [r1 , r2 , r3 ] [1 / 3, 1 / 3, 1 / 3] 30 Rodrigues parameters 0 R r3 r2 ri r3 0 r1 r2 r1 0 1 ri e ijk R jk 2 1 e ijk O jk 1 Oll Oij 1 1 rk rk d ij 2ri rj 2e ijk rk 1 rl rl Rodrigues parameters / unit quaternion: r O Rij e ijk rk qi ri q0 Rodrigues space r ' O' r r ' OO' Wikipedia.com one-to-one corespondence (r r ' ) i ri r 'i e ijk rj r 'k 1 rl r 'l 31 Axis and angle Euler theorem rotation axis magnitude of rotation – rotation angle Or r - Rodrigues vector is parallel to rotation axis The axis is represented by vector n of unit magnitude k n 0 k k 1 k n r r Ok k cos n Ok cos Oii 1 2 Wikipedia.com 32 Axis and angle qi sin( 2) ni ri tan( 2) ni Oij (n, ) d ij cos ni n j (1 cos ) e ijk nk sin O(n, ) O(n,2 ) nS2 0 n S 2 0 2 2 0 0 33 Axis and angle, example 2 / 3 1/ 3 2 / 3 O 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 r [r1 , r2 , r3 ] [1 / 3, 1 / 3, 1 / 3] r n [1 / 3, 1 / 3, 1 / 3 ] [1, 1, 1] / 3 r r cos Oii 1 2 1 / 2 3 60 1 (n, ) [1, 1, 1], 3 3 34 Rotation vector 1 (n, ) [1, 1, 1], 3 3 1 1 r tan( / 2) n tan( / 6) [1, 1, 1] [1, 1, 1] 3 3 n 3 3 [1, 1, 1] f ni i Wikipedia.com 35 Rotation vector 3 i f ni f 0 0 f : [0, ] R strictly increasing, f(w) n 0 i i : 2 1 parametric ball 3 tan( / 2) 2.5 f 2 f( ) 1.5 f tan / 2 sin( ) sin( / 2) 1 0.5 f sin / 2 tan( / 4) [(3 /( 4 2 ))( sin( ))]1/ 3 0 0 f 3 sin / 4 20 Wikipedia.com 2 40 1/ 3 60 80 100 120 140 160 180 in degrees – isochoric parameters 36 Euler angles O O(n, ) x-convention O(1 , , 2 ) O(ez , 1 ) O(e' x , ) O(e"z , 2 ) ey e”z ez ez e”y e”x e’x ex e’x O(1 , , 2 ) O(ez , 2 ) O(ex , ) O(ez , 1 ) 37 Euler angles O(1 , , 2 ) O(ez , 2 ) O(ex , ) O(ez , 1 ) The domain of all proper rotations is covered when 0 1 2 0 0 2 2 1 2 0 1 2 0 0 2 2 38 Euler angles, example O O(n, ) O(1 , , 2 ) O(ez , 2 ) O(ex , ) O(ez , 1 ) 1 90 2 90 60 0 1 0 O(ez ,90) 1 0 0 0 0 1 0 1 O(ex ,60) 0 1/ 2 0 3 / 2 0 0 1 0 1 O(90,60,90) 1 0 0 0 1/ 2 0 0 1 0 3 / 2 (n, ) [1, 0, 0 3 / 2 1 / 2 0 0 1 0 1 / 2 0 3 / 2 1 0 0 0 1 1 / 2 0 0 1 3 / 2 0 3] / 2 , 3 / 2 0 1 / 2 39 Euler angles O(1 , , 2 ) O(ez , 2 ) O(ex , ) O(ez , 1 ) Singularity: Lattman angles: O(1 ,0, 2 ) O(1 ,0, 2 ) O(1 , , 2 ) O(1 , , 2 ) 1 2 2 1 2 2 40 Miller indices e.g., 110001 uvw hkl hu kv lw 0 ei aj – i-th external basis vector – j-th basis vector of crystal direct lattice hkl Ai1Oi3 , Ai 2Oi3 , Ai3Oi3 Cubic system Aij ei a j B A1 uvw B1iOi1, B2iOi1, B3iOi1 hkl O13O23O33 uvw O11O21O31 Wikipedia.com 41 Miller indices, example Cubic system hkl O13O23O33 2 / 3 1/ 3 2 / 3 O 2 / 3 2 / 3 1 / 3 1 / 3 2 / 3 2 / 3 uvw hkl uvw O11O21O31 hkl O13O23O33 (2 1 2) uvw O11O21O31 [2 2 1] hkl uvw (2 1 2)[2 2 1] 42 Summary 43 Summary • Euler theorem: Rotation about a point is equivalent to a rotation about a line. • There is one-to-one correspondence between object's orientations and rotations about a fixed point. • An orientation of an object can be represented an orthogonal matrix. (Proper) rotations are represented (special) orthogonal matrices. Composition of rotations corresponds to multiplication of representing them orthogonal matrices. • There is two-to-one correspondence between unit quaternions and special orthogonal matrices. An orientation of an object can be represented a unit quaterion. Composition of rotations corresponds to multiplication of representing them unit quaternions. 44 Summary • There are a number of orientation parameterizations. „Nice” parameterizations involve more than 3 numbers. There is a relationship between antisymmetric matrices and special orthogonal matrices (Cayley transformation). Cayley transformation is a good starting point for deriving 3-number rotation parameterizations. • Axis and angle • Euler angles • Rodrigues parameters • Rotation vector • Miller indices 45 Unit quaternions and rotations q - components of a unit quaternion i, j, k 1,2,3 0,1,2,3 q q q02 q12 q22 q32 1 Oij q0 qk qk d ij 2qi q j 2e ijk qk q0 2 O – special orthogonal matrix qi e ijk Okj 2 qi e ijk O jk 2 1 Oll 1/ 2 46
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