¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

misorientation in global coordinates by performing the inverse transformation (from
crystal to reference frame) so that ∆a global = a1 T a2.
3.L.6 Effect of Crystal Symmetry on Misorientation
Now that we have established our definitions of misorientation, we can look at the
effect of symmetry operators. As before, we consider axis transformations first. Writing
crystal symmetry operators as O, we can write the general misorientation as follows,
where O1 is the symmetry operator applied to the first crystal, and O2 is the (crystal)
symmetry operator applied to the second crystal.
∆a' = O2 a2 a1 T O1 T (3.M.6.i)
The purpose here is to develop a method of finding all the physically equivalent
relationships between two lattices. [Caution - boundaries require at least two more
parameters in order to account for the inclination of the boundary, making five in total.]
Since the crystal symmetry operators can be regarded as re-labeling the crystal axes,
applying a symmetry operator to either crystal cannot change the physical nature of the
lattice relationship, or, by implication, of the boundary between the two lattices.
3.L.7 Switching Symmetry: Inversion of Rotation
Note that since trace(a) = trace(aT) the rotation angle is invariant with respect to
transposition of the rotation matrix. The rotation axis, however, reverses sign when you
transpose the rotation matrix. Therefore we can write that a( ,θ) = a( ,-θ) T = a(- ,θ) T
and that a( ,-θ) = a(- ,θ), or, a(- ,-θ) = a( ,θ). This result has an important physical
consequence. A given boundary formed as ∆a = a2.a1 T cannot be physically
distinguishable from the boundary formed by the transformation in the reverse sense, i.e.
∆a' ← = → ⎯ a1.a2 T . However, ∆a = (∆a') -1 = (∆a') T r ˆ r ˆ
r ˆ
r ˆ r ˆ
r ˆ r ˆ
. Therefore the misorientations
defined by a( r ˆ ,θ) and a( r
ˆ ,-θ) are physically equivalent. In more physical terms, there is
a switching symmetry, i.e. it does not make any difference to the boundary to switch the
order in which the grains are considered for constructing the misorientation.
This equivalence is useful when determining the fundamental zone in whichever space is
used to describe misorientation. In the Rodrigues space, for example, the application of
the (24 x 24) symmetry operators on both orientations, as in
∆a' = O2 a2.a1 T O1 T , (3.M.7.i)
results in a fundamental zone that contains two unit triangles in terms of rotation
directions. The switching symmetry, i.e. the equivalence of the positive and negative
rotation axes means, however, that the two triangles can be further reduced to a single
triangle. That is to say, two Rodrigues vectors that are equal in magnitude but opposite in
direction represent physically indistinguishable grain boundaries. This is important in the
determination of disorientation, 3.M.9.
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