¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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see below, the application of sample and crystal symmetry permits a reduction in the range of the angles that are required. 3.C.2 Kocks and Roe Euler angles In Kocks, or “symmetric” angles [Kocks, U. F. (1988), “A symmetric set of Euler angles and oblique orientation space”, Eighth International Conference on Textures of Materials (ICOTOM-8), Santa Fe, New Mexico, USA, TMS, Warrendale, Pennsylvania, pp 31-36], the transformation matrix is as follows. The term symmetric refers to the fact that in this convention, the first rotation is positive (anticlockwise) with respect to the local frame, no matter whether one starts with the sample or the crystal frame. This point will become more apparent when we consider graphical representation of orientations. ⎛ −sin Ψsinφ − cosΨcosφ cosθ cos Ψsinφ − sinΨcosφ cosθ cosφ sinθ ⎞ a(Ψ,θ,φ) = ⎜ sinΨ cosφ − cosΨsinφ cosθ ⎝ cos Ψsinθ −cos Ψcosφ − sinΨsinφ cosθ sin Ψsinθ sinφ sinθ ⎟ cosθ ⎠ (3.C.2.1). In Roe angles, the transformation matrix is as follows. a(ψ,θ,φ) = ⎛ −sinψ sinφ + cosψ cosφ cosθ cosψ sinφ + sinψ cosφ cosθ −cosφ sinθ⎞ ⎜ −sinψ cosφ − cosψ sinφ cosθ ⎝ cosψ sinθ cosψ cosφ − sinψ sinφ cosθ sinψ sinθ sinφ sinθ ⎟ cosθ ⎠ (3.C.2.2). 8/27/09 10
Fig. 3.C.2. Kocks Euler angles illustrated by analogy to the position and the heading of a boat with respect to the globe. Latitude (Θ) and longitude (ψ) describe the position of the boat; third angle describes the heading (φ) of the boat relative to the line of longitude that connects the boat to the North Pole. The following table describes the different conventions for Euler angles and the way in which they area related. Table 3.1. Definitions of Euler Angles and Conversions Convention 1st 2nd 3rd 2nd angle about axis: Kocks (symmetric) Ψ Θ φ y Bunge Matthies φ1-π/2 α Φ β π/2−φ2 π−γ x y Roe Ψ Θ π−Φ y Fig. 3.C.3. Diagrams illustrating the differences between Kocks, Roe, Bunge and Canova Euler angle definitions. The first row, (a), shows pole figures (crystal z axis in relation to sample axes) and the second row, (b), shows inverse pole figures (sample Z in relation to crystal axes). Bunge and Canova are inverse to one another Kocks and Roe differ by sign of third angle Bunge rotates about x’, Roe/Kocks about y’ (2nd angle rotation). An important aspect of Euler angle representation is that when the second angle is exactly zero, then the other two angles can be combined together, i.e. there is a singularity near the origin of the space. This leads to the idea that there may be circumstances under which it is helpful to use the sum and difference of the first and third angles. These were introduced by Bunge [(1988). Calculation and representation of the complete ODF. ICOTOM-8, Santa Fe, TMS, Warrendale, PA.] and Helming et al. [Helming, K., S. Matthies, et al. (1988). ODF representation by means of sigma sections. 8/27/09 11
Page 1 and 2: 3. Mathematical Representation of C
Page 3 and 4: 3.A Orthogonal Tensors of Rank 2. R
Page 5 and 6: Fig. 3.B.1. Diagram showing the rel
Page 7 and 8: 3.C.1 Bunge Euler angles We will us
Page 9: Figure 3.C.1 Diagrams showing the s
Page 13 and 14: If the second angle is near to zero
Page 15 and 16: esult that all vectors correspondin
Page 17 and 18: sinθ 2 = 1 2 3− tr a ( ) (3.F.6)
Page 19 and 20: xi = (q4 2 -q1 2 -q2 2 -q3 2 )Xi +
Page 21 and 22: Fig. Gii.2. Schematic {100} pole fi
Page 23 and 24: Fig. Gii.4. Schematic {110} pole fi
Page 25 and 26: physically distinguishable objects.
Page 27 and 28: ⎛ −1 0 0⎞ ⎜ ⎟ ⎜ 0 1 0
Page 29 and 30: 3.K Effect of Symmetry on Orientati
Page 31 and 32: ∆a = a2.a1 T (3.M.3.i) Note that
Page 33 and 34: 3.L.9 Disorientation The Disorienta
Page 35 and 36: 3.L.11 Misorientation (Active Rotat
Page 37 and 38: 3.L.13 Range of Parameters for Rodr
Page 39 and 40: Now, at last, we can apply this bis
Page 41 and 42: q’=q•S, where S is the symmetry
Page 43 and 44: To project the point w onto the pag
Page 45 and 46: Pole Figure Orientation Distributio
Page 47 and 48: ( ) = f h,ζ,w P h w ζ = 2π ∫

¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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