¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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If we write the OD as f(g) in the conventional manner, then we can calculate the intensity in the pole figure, P(w), by summing over all the symmetry operators and integrating over the range of Euler angles (also a sum if the OD is in discrete form). Note that the coordinates in the pole figure are determined by the relation above. More specifically, the three Euler angles in the expression below are not independent of one another and can be written in terms of a single parameter. The functional relationships are to be determined from the previous expression, i.e. that which relates the point, w, in pole figure space to a line in orientation space. ( ( ) ) ∑ P(w) = f a −1 ∫∫∫ ψ ,Θ,φ sinΘdψdΘdφ S (k) h 8/27/09 44 k
Pole Figure Orientation Distribution (#,$,") 1 (#,$,") 2 (!,") (#,$,") 3 (#,$,") 4 (#,$,") 5 To make this approach more concrete and specific, consider a way in which this relationship can be re-cast in terms of a single integration parameter. Start with the crystal in the reference configuration and think in terms of active rotations (vector transformations), i.e. we work in a fixed frame of reference corresponding to that used for sample space (in which the pole figure is plotted). We first rotate the crystal with th=th( t ˆ h ,θ) so as to place the pole of interest, h, in the center of the hemisphere, i.e. at the North pole. This matrix can be obtained by first finding a rotation axis, t ˆ h , as the vector product of [001] and h. ˆ t h = [001] ! h [001] ! h = h2 2 ["h2,h 1,0] 2 ) + h1 The rotation angle, θ, is given by the inverse cosine of the scalar product: cos! = h 3 From this axis-angle pair, the rotation matrix can be constructed as follows. 8/27/09 45
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 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: To project the point w onto the pag
Page 47 and 48: ( ) = f h,ζ,w P h w ζ = 2π ∫

¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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