¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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ICOTOM-8, Santa Fe, TMS, Warrendale, PA.] to address this issue. The first and third angles can be replaced by the following, where ν is equivalent to a parameter β used previously by Williams. ν = (Ψ-φ)/2 µ = (Ψ+φ)/2 (3.C.2.3) These are related to the oblique section parameters of Bunge (φ + ,φ - ) and Helming (σ,δ) by φ + = σ = ν+π/2, and φ - = δ+π/2 = µ. Note that when two rotations are expressed in the form of Euler angles, there is no simple method for combining them. It is usually sensible to convert them to matrix form before performing manipulations on them. The action of symmetry operators on orientations can, however, be expressed in reasonably simple terms. 3.C.3 Conversions To convert from Miller index description to Euler angles, one may use these formulae. ϕ 1 = sin −1 Φ = cos −1 l h 2 + k 2 + l 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ ϕ2 = cos −1 k h 2 + k 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ w u 2 + v 2 + w 2 h 2 + k 2 + l 2 h 2 + k 2 ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ (3.C.3.1) A limitation of these formulae from a numerical perspective is the range of inverse trigonometric functions. The only such function that can cover the range 0-2π is the inverse tangent € function with two arguments, generally written as ATAN2 (lower case for C programs). Considerable care with this inverse trigonometric function is required because in some software packages such as Excel, the two arguments are ordered as {x,y}, whereas in C and Fortran, they are ordered in the opposite fashion as {y,x}. Also, if both arguments approach zero, the result is undefined and the function will return an error. This means that the programmer has to detect this condition and take special measures, which can easily happen for orientations close to the reference position (cube component). Therefore it is better to write the orientation (rotation) matrix in terms of the Miller indices, and then convert from matrix to Euler angles, as given here: € ( ) Φ = cos −1 a33 ϕ 2 = tan −1⎛ ( a13 ⎜ ⎝ sin Φ) ( a23 ⎞ sin Φ ⎟ ) ⎠ ϕ1 = tan −1⎛ ( a31 sin Φ) ⎜ ⎝ ( −a32 ⎞ sin Φ ⎟ ) ⎠ (3.C.3.2) 8/27/09 12
If the second angle is near to zero or to 180°, then these formulae become inaccurate (or unusable when it is exactly equal to zero). In this circumstance, the following formulae may be used. if a 33 ≈1, Φ = 0, ϕ 1 = tan −1 a ⎛ ⎜ 12 ⎝ 2 about what € values to assign to the first and third angles. a 11 ⎞ ⎟ ⎠ , and ϕ2 = ϕ1 (3.C.3.3) There is one last special case, which is for orientations (symmetrically related to the exact cube) where a11 and a12 are both close to zero. In this an arbitrary decision must be made 3.D Axis-angle pair representation Another very useful description of rotation is in terms of a single rotation axis and an associated rotation angle where the location of the axis is arbitrary (as opposed to axes that are fixed on one of the reference axes). This is not used for orientation (texture) because it reveals little of interest about the nature of the orientation. It is much used for grain boundaries, however, because the magnitude of the rotation is the primary means of distinguishing low angle boundaries from high angle boundaries. Note that this method still only requires three independent parameters because the axis represents two parameters (three direction cosines constrained to be a unit vector) and the angle is a third parameter. This is commonly written as ( r ˆ ,θ ) or as (n,ω). The following figure illustrates the effect of a rotation about an arbitrary axis, OQ (equivalent to r ˆ and n) through an angle α (equivalent to θ and ω). Figure 3.D.2 Showing rotation about an arbitrary axis OQ through α. After Spencer [Spencer, A. J. M. (1980). Continuum Mechanics. New York, Longman.] 8/27/09 13
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: Fig. 3.C.2. Kocks Euler angles illu
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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