¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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4) 3) 2) 1) B B A A ns ^ A !g BA ns ^ B B ns ^ gB ns ^ !g A BA ns +gB ns ^ -gA ns ^ !g AB -gB ns ^ gA ns ^ !g AB Figure 3.L.1. Showing the four physically equivalent variants of a grain boundary description involving misorientation and inclination. The order in which the misorientation is constructed can be switched and the sense of the boundary normal can be switched (provided that the crystals are centro-symmetric). ^ -gB ns ^ gA ns ^ -gA ns ^ 8/27/09 34
3.L.11 Misorientation (Active Rotation) and (Crystal) Symmetry Let us now expand the active rotation description of misorientation to include symmetry operators, confining our attention to crystal symmetries. The form of the equation becomes the following, where γ is a rotation drawn from the set of rotation operators in the group of crystal symmetries (following the notation of Adams et al.). ∆g' = g γ (g γ ) 2 2 1 1 T = g γ γ T T g1 (3.M.10.i) 2 2 1 Note the different order of applying the symmetry operators as compared with Eq. 11.6 above. Note that the two symmetry operators can be combined to form a single operator (assuming that we are dealing with a single-phase material) because they are both elements of the same symmetry group. Thus the multiplicity of physically equivalent operators is only 24 x 2 = 48 for misorientations described in the global frame (for cubic crystals), in contrast to the 1152 variants available for misorientations described in the local frame (section 3.L.9 above). If we now change to the local coordinate frame (i.e. that of the first crystal) we obtain a result that looks similar to the axis transformation version (except that the transposed rotation is in the first position). ! g ˜ " 3.L.12 Symmetry Operators in Rodrigues Space = γ1 T g1 T g2 γ2 (3.M.10.ii) The action of symmetry operators in Rodrigues space is derived from consideration of the combination of Rodrigues vectors. The O(432) group can represented by Rodrigues vectors as follows. Note that for 2-fold axes with 180° rotations, tan(θ/2) = tan(π) = ∞. Table 3.3. Symmetry Operators Expressed as Rodrigues Vectors Symmetry Operator Rodrigues Vector 2-fold on 2 L100 ∞(1,0,0) 4-fold on 2-fold on 4 L100 2 L110 (1,0,0) ∞(1,1,0) 3-fold on 3 L111 (1,1,1) These operators immediately illustrate one of the challenges in working with the Rodrigues space because it extends to infinity, at least in principle. In practice, however, for any material possessing more than triclinic symmetry, we can confine our attention to a box –1 ≤ x ≤ 1, −1 ≤ y ≤ 1, −1 ≤ z ≤ 1. The aspect that is more bizarre is the way in which symmetry related points are connected by their symmetry operator. Sutton & Balluffi discuss the example of the 2-fold axis which connects points close to the x=-1 plane to points close to the x = +1 plane, but with a 90° rotation. Frank likens this to a Moebius strip-like connection. 8/27/09 35
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 and 10: Figure 3.C.1 Diagrams showing the s
Page 11 and 12: Fig. 3.C.2. Kocks Euler angles illu
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: 3.L.9 Disorientation The Disorienta
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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