¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

3.L Effect of Symmetry in Rodrigues space, with special reference to Misorientations,
or, how to get from one crystal to another (not necessarily adjacent):
3.L.1 Outline
We explain the difference between forming a misorientation from passive
transformations ( = ab•aa -1 ) which is (automatically) in a crystal frame, and forming
it as a rotation from crystal A to crystal B (∆g = gb•ga -1 ! g ˜
). Note that inverting the
misorientation (i.e. go from B to A) does not change the magnitude or the axis of
rotation, except that the axis is inverted (equivalent to reversing the sign of rotation: the
standard formulae effectively invert the axis, however). The latter (rotation) is in the
global reference frame, so you have to transform its axes to get the local frame,
! g
˜
= aa -1 • ∆g • aa = ga -1 •gb.
3.L.2 Active Rotations versus Axis Transformations (Passive Rotations)
Once again we must deal with the two complementary ways of considering texture that
are available to us. Recall that the standard texts (Bunge, Kocks/Tomé/Wenk) assume
that orientations are defined as transformations of axes. They provide the standard
definition wherein each coefficient of the transformation matrix, a, is a direction cosine
between the appropriate pair of new (crystal) and old (reference) basis (unit) vectors, e',
e, respectively.
aij = ei' • ej (3.M.2.i)
The standard texts typically use the notation g, for this same matrix but we reserve this
notation for the matrix that describes the active rotation operator customary in the
writings of Adams and his collaborators. As described above, the matrix that describes
the same active rotation (assuming orthogonal basis vectors) is simply the transpose of
the equivalent passive rotation. Thus if we describe the orientation of a grain in terms of
the transformation required to express a tensor quantity in grain (crystal) axes that is
given in terms of the reference (sample) axes with the matrix a, then the rotation that
would physically rotate the same grain from the reference configuration (the "cube"
component) to its actual position in space (neglecting translations) is g, where g = a T .
All this review may seem somewhat overly detailed but it is essential in order to avoid
confusion when comparing the derivations of different authors.
3.L.3 Misorientation (Axis Transformation)
Consider the misorientation to be the transformation that converts a tensor
quantity expressed in one set of crystal coordinates to those of another crystal. One can
think of two successive transformations where the first converts from the first crystal to
the reference frame, a1 T , and the second converts from the reference frame to the second
crystal, a2. Maintaining the definition for transformation matrices given above, the
misorientation, ∆a, is given by the following.
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