¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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Fig. 3.J.2: table (from Kocks et al.) of symmetry operators expressed as orthogonal matrices. Note that the a33 entry for the 1 st column, third matrix down (with “C2” against it) should have a minus sign (to make it -1). 8/27/09 28
3.K Effect of Symmetry on Orientation, especially in Euler-angle space The effect of symmetry on orientations can now be treated by reference to specific point groups. For cubic materials, we use the O(432) point group for the crystal symmetry. Similarly for the orthotropic sample symmetry typically observed in plane strain compression we can use the mmm point group which contains three orthogonal mirrors. This combination of mirrors is special in the sense that an inversion center is also present which means that we can substitute three orthogonal diads for the mirrors and work with proper rotations only. The effect of the symmetry elements on the Euler angles varies in complexity with the element. A diad on the (crystal) z-axis relates g(φ 1,Φ,φ 2) to g(φ 1,Φ,φ 2+π), for example. A triad on [111] has a much more complex effect, however. Rather than explore this in detail, the reader is directed towards more detailed discussion in Bunge, who in turn refers to Pospiech [Pospiech, J., A. Gnatek, and K. Fichtner, Symmetry in the space of Euler angles. Kristall und Technik, 1974. 4: p. 729.]. The 3-fold axis introduces a screwaxis (by analogy) into the Euler space. The following table provides a partial listing of the symmetry elements and their effect on Euler angles. Table 3.2. Effect of Symmetry Elements on Euler Angles Symmetry Element Bunge Symm -etric (Kocks) 2-fold axis on Sample X π-φ1 π-Φ φ2±π −Ψ π-Θ φ±π 2-fold axis on Sample Y 2-fold axis on Sample Z -φ1 φ1-π π-Φ Φ φ2±π φ2 π-Ψ Ψ±π π-Θ Θ φ±π φ 2-fold axis on Crystal x φ1±π π-Φ π-φ2 Ψ±π π-Θ -φ 2-fold axis on Crystal y 2-fold axis on Crystal z 3-fold axis on Crystal z 4-fold axis on Crystal z 6-fold axis on Crystal z φ1±π φ1 φ1 φ1 φ1 π-Φ Φ Φ Φ Φ −φ2 φ2±π φ2±2π/3 φ2±π/2 φ2±π/3 Ψ±π Ψ Ψ Ψ Ψ π-Θ Θ Θ Θ Θ π-φ φ±π φ±2π/3 φ±π/2 φ±π/3 The range of Euler angles used for representing textures depends, obviously on the combination of symmetry subgroups appropriate for the systems of interest. Although the cubic-orthorhombic combination (i.e. O(222) for sample symmetry, O(432) for crystal symmetry) would permit a much smaller fundamental zone to be used, the actual choice is typically 0≤φ 1≤90°, 0≤Φ≤90°, and 0≤φ 2≤90° with exactly the same ranges used for any choice of Euler angle convention. The results in the subspace being three times larger than the fundamental zone; the 3-fold axis that is not exploited produces peculiar effects, however, in terms of which points are related to which others, as noted above. More specifically, the number of physically equivalent copies of an orientation for cubic/orthorhombic symmetry in any space is 4 x 24 = 96. Restricting the ranges of the Euler angles as above means a reduction in the space of 4 x 2 x 4 = 32: the missing factor of 3 is exactly the consequence of neglecting the triad symmetry element. 8/27/09 29
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: ⎛ −1 0 0⎞ ⎜ ⎟ ⎜ 0 1 0
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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