Crystallographic orienta1on representa1ons

Transcription

1 Crystallographic orienta1on representa1ons -- Euler Angles -- Axis-Angle -- Rodrigues-Frank Vectors -- Unit Quaternions Tugce Ozturk, Anthony Rolle Texture, Microstructure & Anisotropy April 1, 016 0

2 Lecture ObjecJves Brief review on the challenges of describing crystallographic orientajons Review of previously studied orientajon representajons: rotajon matrices and Euler angles IntroducJon of Rodrigues-Frank vectors IntroducJon of unit uaternions Useful math for using uaternions as rotajon operators Conversions between Euler angles, axis-angle pairs, RF vectors, and uaternions 1

3 Why can it get challenging to work with crystallographic orientajons? OrientaJon is specified based on reference frames Sample coordinates Crystal coordinates Lab coordinates Different choices of axes to convert crystal frames Hexagonal frame into orthonormal frame X= [10 10] Y= [1 10] Z= [0001] or X= [ 1 10] Y= [01 10] Z= [0001] Personal experiences?

4 Descriptors of orientajon OrientaJon matrix MisorientaJon matrix Miller indices Euler angles Axis/angle pair Rodrigues Vectors Unit Quaternions Pole Figure Inverse PF Euler space ODF and modf Axis/angle space ODF and modf Rodrigues space ODF and modf Quaternion space (4D) ODF and modf ComputaJonal efficiency 3

5 Crystal OrientaJon: RotaJon (direcjon cosine) Matrices Crystallographic orientajon is the posijon of crystal system with respect to the specimen coordinate system. A single rotajon can be described by a rotajon axis and an angular offset about the axis. Passive rota+on: RotaJon of axes, sample coordinate into crystal coordinate. AcJve rotajon: RotaJon of the object, crystal coordinate into sample coordinate. R = cosθ sinθ sinθ cosθ vs. R = cosθ sinθ sinθ cosθ 3x3 rotajon matrices, R, leh-muljply column vectors ProperJes: R -1 = R T and det (R) = 1 Rows and columns are unit vectors, and the cross product of two rows or columns gives the third. 4

6 Crystal OrientaJon: Miller Indices NotaJon Three orthogonal direcjons chosen as the reference frame All direcjons can then be described as linear combinajons of the three unit direcjon vectors. Most straighkorward in orthonormal systems, e.g. cubic, as indices for plane and direcjon are idenjcal. In many cases we use the metallurgical names Rolling DirecJon (RD) // x, Transverse DirecJon (TD) // y, and Normal DirecJon (ND) // z. We then idenjfy a plane normal parallel to 3 rd axis (ND) and a crystal direcjon parallel to the 1 st axis (RD), wri4en as (hkl)[uvw]. (hkl) Z Z Y X [uvw] X 5

7 Crystal OrientaJon: Euler Angles Euler showed three seuenjal rotajons about different axes can describe orientajons (18 th century) Passive rotajon (SCoord into CCoord) There are 1 different possible axis-angle seuences. The standard seuence varies from field to field. MulJple convenjons Proper Euler angles (z-x-z, x-y-x, y-z-y, z-y-z, x-z-x, y-x-y) Tait Bryan angles (x-y-z, y-z-x, z-x-y, x-z-y, z-y-x, y-x-z) We use the Bunge convenjon. First angle φ 1 about ND Second angle Φ about new RD Third angle φ about newest ND Sample symmetry effects the size of Euler space Most general, triclinic crystal system, no sample symmetry à 0 φ 1, φ Φ 180 Cubic crystal system, rolled sample (orthonormal sample symmetry) à 0 φ 1, φ Φ 90 6

8 RotaJon matrices from Miller indices [uvw] RD TD ND (hkl) [100] direcjon [010] direcjon [001] direcjon ˆ n = (h, k, l) h + k + l a 11 a 1 a 13 a 1 a a 3 a 31 a 3 a 33 a ij = Crystal ˆ b = Rows are the direcjon cosines for [100], [010], and [001] in the sample coordinate system, columns are the direcjon cosines (i.e. hkl or uvw) for the RD, TD and ND in the crystal coordinate system. (u, v, w) u + v + w Sample b 1 t 1 n 1 b t n b 3 t 3 n 3 ˆ n t = ˆ b ˆ n ˆ b ˆ Challenge: With Miller indices, we define the closest family (can be degrees away) 7

9 RotaJon matrices from Bunge Euler angles $ cosφ 1 sinφ 1 0' $ ' & ) Z 1 = sinφ 1 cosφ 1 0 & ), X = & ) 0 cosφ sinφ & ) % 0 0 1( % 0 sinφ cosφ( $ cosφ sinφ 0' & ) Z = sinφ cosφ 0 & ) % 0 0 1( A = Z XZ 1 = Combine the 3 rotajons via matrix muljplicajon; 1 st on the right, last on the leh: [uvw] $ cosϕ 1 cosϕ & sinϕ 1 sinϕ cosφ & & & cosϕ 1 sinϕ & & sinϕ 1 cosϕ cosφ & & % sinϕ 1 cosϕ +cosϕ 1 sinϕ cosφ (hkl) sinϕ sin Φ ' ) ) ) sinϕ 1 sinϕ cosϕ sin Φ) ) +cosϕ 1 cosϕ cosφ ) ) ) sinϕ 1 sinφ cosϕ 1 sinφ cosφ ( 8

10 Miller indices from Bunge Euler angles Compare the indices matrix from slide 6 with the Euler angle matrix from slide 8. u = v = h = nsin Φsinϕ k = nsin Φcosϕ l = ncosφ n ʹ cosϕ 1 cosϕ sinϕ 1 sinϕ cos Φ ( ) ( ) n ʹ cosϕ 1 sinϕ sinϕ 1 cosϕ cos Φ w = n ʹ sinφ sinϕ 1 n, n = factors to make integers towards the closest family 9

11 Bunge Euler angles from rotajon matrices Notes: The range of inverse cosine (ACOS) is 0-π, which is sufficient for Φ; from this, sin(φ) can be obtained; The range of inverse tangent is 0-π, (must use the ATAN funcjon) which is reuired for calculajng φ 1 and φ. Φ = cos 1 a 33 ϕ = tan 1 ( a 13 sinφ) ϕ 1 = tan 1 ( a 31 sinφ) ( ) a 3 sinφ sinφ ( ) a 3 ( ) if a 33 1, Φ = 0, ϕ 1 = tan 1 a 1 a11, and ϕ = ϕ 1 10

12 Bunge Euler angles from miller indices l cos Φ = h + k + l k cosϕ = h + k w h sinϕ 1 = + k + l u + v + w h + k Caution: when one uses the inverse trig functions, the range of result is limited to 0 cos -1 θ 180, or -90 sin -1 θ 90. Thus it is not possible to access the full range of the angles. It is more reliable to go from Miller indices to an orientation matrix, and then calculate the Euler angles. Extra credit: show that the following surmise is correct. If a plane, hkl, is chosen in the lower hemisphere, l<0, show that the Euler angles are incorrect. 11

13 Crystal OrientaJon: Axis-angle Pair Axis-angle pair is defined such that the crystal frame and the sample frame is overlapped by a single rotajon, θ, along a common axis [u v w]. Commonly wri4en as ( r ˆ,θ) or (n,ω) AcJve rotajon The rotajon can be converted to a matrix (passive rotajon) Common in misorientajon representajon - Difference of misorientajon vs. rotajon difference is the choice of reference axis Easy conversion into Rodrigues-Frank vectors 1

14 For( r ˆ,θ) or (n,ω) RotaJon matrices from Axis-angle g ij = δ ij cosθ + r i r j ( 1 cosθ) k=1,3 + ε ijk r k sinθ ( ) uv( 1 cosθ) + wsinθ uw( 1 cosθ) v sinθ ( ) wsinθ cosθ + v ( 1 cosθ) vw( 1 cosθ) + usinθ ( ) + v sinθ vw( 1 cosθ) usinθ cosθ + w ( 1 cosθ) cosθ + u 1 cosθ = uv 1 cosθ uw 1 cosθ This form of the rotation matrix is a passive rotation, appropriate to axis transformations 13

15 RotaJon axis, n: Axis-Angle from RotaJon matrices (a a ), (a a ), (a a ) n = (a a ) + (a a ) + (a a ) Note the order (very important) of the coefficients in each subtracjon; again, if the matrix represents an ac1ve rota1on, then the sign is inverted. RotaJon angle, θ: a ii = 3cosθ + (1 cosθ)n i =1+ cosθ therefore, cos θ = 0.5 (trace(a) 1). 14

16 Crystal OrientaJon: Rodrigues-Frank Vectors Rodrigues vectors were popularized by Frank [ OrientaJon mapping. Metall. Trans. 19A: (1988)], hence the term Rodrigues-Frank space Easily converted from axis-angle Most useful for representajon of misorienta1ons, also useful for orientajons The RF representajon from axis angle R(r, α ) scales r by the tangent of α/ ρ = ˆr tan( α / ) Note semi-angle 15

17 Rodrigues-Frank Space PopulaJon of vectors Smallest R results from the smallest θ, and it lies closest to the origin of the RF space Lines in RF space represent rotajon about fixed axis Further, when the RF space is divided into subvolumes, the misorientajons that lie on the boundary of the subvolume are geometrically special, by having a low index axis of misorientajon. As the Rodrigues vector is so closely linked to the rotajon axis, which is meaningful for the crystallography of grain boundaries, the RF representajon of misorientajons is very common. Cubic crystal symmetry no sample symmetry Cubic orthorhombic Cubic cubic 16

18 R-F Vector from RotaJon matrices Simple formula, due to Morawiec: ρ 1 ρ ρ 3 = (a 3 a 3 ) / 1+ tr(a) (a 31 a 13 ) / 1+ tr(a) (a 1 a 1 ) / 1+ tr(a) [ ] [ ] [ ] Trace of a matrix: tr(a) = a 11 + a + a 33 17

19 R-F Vector from Bunge Euler angles and Bunge Euler from R-F vectors tan(α/) = {(1/[cos(Φ/) cos{(φ 1 + φ )/}] 1} ρ 1 = tan(φ/) [cos{(φ 1 - φ )/}/cos{(φ 1 + φ )/}] ρ = tan(φ/) [sin{(φ 1 - φ )/}/[cos{(φ 1 + φ )/}] ρ 3 = tan{(φ 1 + φ )/} P. Neumann (1991). Representation of orientations of symmetrical objects by Rodrigues vectors. Textures and Microstructures 14-18: # ## Conversion from Rodrigues to Bunge Euler angles: sum = atan(r 3 ) ; diff = atan ( R /R 1 ) φ 1 = sum + diff; Φ =. * atan(r * cos(sum) / sin(diff) ); φ = sum - diff 18

20 RotaJon matrix from R-F vectors Due to Morawiec: Example for the 1 entry: NB. Morawiec s E. on p. has a minus sign in front of the last term; this will give an acjve rotajon matrix, rather than the passive rotajon matrix seen here. 19

21 Crystal OrientaJon: Unit Quaternions What is a uaternion? A uaternion (1843 by Hamilton) is an ordered set of four real numbers 0, 1,, and 4. Here, i, j, k are the familiar unit vectors that correspond to the x-, y-, and z-axes, resp. Scalar part Vector part [1] Quaternion muljplicajon is non-commutajve (p p) [] An extension to D complex numbers [3] Of the 4 components, one is a real scalar number, and the other 3 form a vector in imaginary ijk space! = + i i j = 0 + i1 + j k3 = j = k = ijk = 1 jk = kj = ki = ik k = ij = ji 0

22 Quaternions as rotajons A uaternion can represent a rotajon by an angle θ around a unit axis n= (x y z): = cos θ xsin θ ysin θ zsin θ or = cos θ, nsinθ Checking the magnitude of the uaternion, = Unit! = cos θ + x sin θ + y sin θ + z sin θ ( ) = cos θ + θ sin x + y + x = cos θ + θ sin n = cos θ + θ sin = 1 =1 1

23 Unit Quaternion from RotaJon Matrices Formulae, due to Morawiec: Note: passive rotajon/ axis transformajon (axis changes sign for for acjve rotajon) cosθ = 1 1+ tr Δg ( ) 4 = ± i = ± ε ijk Δg jk ( ) 4 1+ tr Δg 1+ tr( Δg) ±[Δg(,3) Δg(3,)]/ 1+ tr Δg ±[Δg(3,1) Δg(1,3)]/ 1+ tr Δg = ±[Δg(1,) Δg(,1)]/ 1+ tr Δg ± 1+ tr( Δg) / Note the coordinajon of choice of sign! ( ) ( ) ( )

24 3 RotaJon Matrices from Unit Quaternions g= For OrientaJon matrix: Quaternions are ComputaJonally efficient muljplicajon reuires fewer computajons Axis-angle and R-F into uaternion conversion is simple

25 RotaJon of a vector by a uaternion Passive rotajon AcJve rotajon Consider rotajng the vector i by an angle of α = π/3 about the <111> direcjon. Rotation axis: k j For an active rotation: i For a passive rotation: 4

26 References Frank, F. (1988). OrientaJon mapping, Metallurgical Transac1ons 19A: P. Neumann (1991). Representation of orientations of symmetrical objects by Rodrigues vectors. Textures and Microstructures 14-18: # Takahashi Y, Miyazawa K, Mori M, Ishida Y. (1986). Quaternion representajon of the orientajon relajonship and its applicajon to grain boundary problems. JIMIS-4, pp Minakami, Japan: Trans. Japan Inst. Metals. (1st reference to uaternions to describe grain boundaries). A. Su4on and R. Balluffi (1996), Interfaces in Crystalline Materials, Oxford. V. Randle & O. Engler (010). Texture Analysis: Macrotexture, Microtexture & Orienta1on Mapping. Amsterdam, Holland, Gordon & Breach. S. Altmann (005 - reissue by Dover), Rota1ons, Quaternions and Double Groups, Oxford. A. Morawiec (003), Orienta1ons and Rota1ons, Springer (Europe). On a New Species of Imaginary QuanJJes Connected with a Theory of Quaternions, by William Rowan Hamilton, Proceedings of the Royal Irish Academy, (1844), Des lois géométriues ui régissent les déplacements d un système solide dans l espace et de la variajon des coordonnées provenant de ces déplacements considérées indépendamment des causes ui peuvent les produire, M. Olinde Rodrigues, Journal des MathémaJues Pures et Appliuées,

27 ProperJes of Quaternions Magnitude of a uaternion: Conjugate of a uaternion: AddiJon of uaternions: Product of two arbitrary uaternions Scalar part Using more compact notajon: again, note the + in front of the vector product On a New Species of Imaginary Quantities Connected with a Theory of Quaternions, by William Rowan Hamilton, Proceedings of the Royal Irish Academy, (1844), Vector part 6

28 Cubic Crystal Symmetry Operators The numerical values of these symmetry operators can be found at: h4p://pajarito.materials.cmu.edu/rolle4/texture_subroujnes: uat.cubic.symm etc. 7

29 Combining RotaJons as RF vectors Two Rodrigues vectors combine to form a third, ρ C, as follows, where ρ B follows aher ρ A. Note that this is not the parallelogram law for vectors! ρ C = (ρ A, ρ B ) = {ρ A + ρ B - ρ A x ρ B }/{1 - ρ A ρ B } addition vector product scalar product 8