¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

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The associated (110) pole figure is very similar to the Goss texture pole figure except that it is rotated about the ND. In this example, the crystal has been rotated in only one sense (anticlockwise). Fig. Gii.6. Schematic {001}, {111} and {110} pole figures, reading from left to right, for the Brass component. The RD is vertical in these diagrams and the TD horizontal. Only one variant of the Brass component is shown or clarity; the other variant (related by sample symmetry) can be obtained by a rotation of 70° about the ND (for example). The brass component is convenient because we can think about performing two successive rotations: 1st about the ND, 2nd about the new position of the [100] axis. 1st rotation is 35° about the ND; 2nd rotation is 45° about the [100]. (φ 1, Φ, φ 2) = ( 35°, 45° , 0° ). The existence of variants of a given texture component is a consequence of (statistical) sample symmetry. If one permutes the Miller indices for a given component (for cubic materials, one can change the sign and order, but not the set of digits), then different values of the Euler angles are found for each permutation. If a pole figure is plotted of all the variants, one observes a number of physically distinct orientations, which are related to each other by symmetry operators (diads, typically) fixed in the sample frame of reference. Each physically distinct orientation is a “variant”. The number of variants listed depends on the choice of size of Euler space and the alignment of the component with respect to the sample symmetry. A typical size of Euler space is 90x90x90°; this is appropriate to a combination of cubic crystal symmetry and orthorhombic sample symmetry. The space contains three copies, however, of the fundamental zone, which leads to considerable potential for confusion, as will be discussed in detail below. 3.H Fundamental Zones: Summary What is a “fundamental zone” or “irreducible space”? It is a set of physically distinct orientations (or misorientations) and it is intimately linked to symmetry, both in the crystal and in the sample. This approach illustrates that it is linked to representation of physical objects, whether grains or grain boundaries (or interfaces….). In terms of the mathematical representation, it is the range of the parameters the delineates the zone. In graphical terms, it is the smallest region of the parameter space that encloses all of the 8/27/09 24
physically distinguishable objects. As an example of a familiar fundamental zone, consider the stereographic projection of the holosymmetric point group, m3m, in standard position. We speak of the unit triangle for plane normals because if we restrict the range of Miller indices to h ≥ k ≥ l ≥ 0, then we have delineated all of the different kinds of crystallographic plane. 3.H Fundamental Zones: Orientations A fundamental zone is the range of the space used to describe orientations (or any quantity based on rotational relationships) constructed such that it contains one and only one copy for each physically distinct orientation. Defining fundamental zones is impractical without first discussing symmetry, which follows in the next section of this chapter. It should be intuitively clear that symmetry results in multiple fundamental zones within the overall orientation space. Moreover, since each copy of version of the zone contains exactly the same number of points, the volume associated with each one is identically the same. The fundamental zone in for orientations is generally defined in Euler angle space because that is the space in which the practitioners of texture analysis are familiar with. Nevertheless, one can always define a fundamental zone in any space associated with the parameterization, be it Rodrigues vectors, unit quaternions etc. The awkward feature of Euler angles is that the actual fundamental zone for cubic materials has a rather odd shape, which does not lend itself to making attractive graphics. Therefore the convention is to use Cartesian axes and show several copies of the fundamental zone in any given plot. This is discussed in more detail in the next chapter on graphical representations of texture. 3.H Fundamental Zones: Misorientations Just as the representation of orientation has an irreducible set of orientations, each of which represents a physically distinct relationship to the reference configuration, so too there is a fundamental zone for misorientations. In the parlance adopted above, the fundamental zone contains the entire set of physically distinct disorientations. The shape of the zone depends on the representation chosen. Since an (mis-)orientation matrix has nine coefficients, there is no simple geometrical shape available. For now, we will duck the issue of how to develop the shape in the various representation spaces. Suffice it to say that the shape of the fundamental zone for misorientation is usually defined (and plotted) in R-F space because the boundaries are planes for all the Laue groups. The fundamental zone is delimited by the following inequalities, where ρ = [ρ1,ρ2,ρ3]. ρ1 > ρ2 > ρ3 > 0 0 ≤ ρ1 ≤ (√2-1) ρ2 ≤ ρ1 ρ3 ≤ ρ2 ρ1 + ρ2 + ρ3 ≤ 1 8/27/09 25
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: Fig. Gii.4. Schematic {110} pole fi
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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