¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

It is useful to collect together the various types of representation of orientation or
misorientation. In this context there is no difference between orientation and
misorientation! Both these quantities are represented as rotations and all the various
representation methods are useful, depending upon the problem to be solved. Although
Euler angles are conventionally used to describe textures, and Rodrigues vectors are
typically used to describe grain boundary textures, one can use any representation of
rotations for either of these quantities. Historically, Euler angles were favored because of
the availability of series expansions (in terms of generalized spherical harmonics) in this
space that had long ago been devised for the representation of electron orbitals. Recently,
however, Schuh [Mason, J. K. and C. A. Schuh (2008). “Hyperspherical harmonics for
the representation of crystallographic texture”, Acta materialia 56 6141-6155] has
published an account of hyperspherical harmonics that can be used with the quaternion
representation. Of course, the physical entities represented are different: grains and the
boundaries between them (or different phases) are physically different things. It is also
very useful to keep in mind that an Orientation Distribution, for example, is really a
probability distribution because it tells you the probability of finding a grain within a
certain range of orientation. Be careful, however, of the distributions used in materials
science because they are not, strictly speaking probability densities because they are
normalized to have unit average density, regardless of the parameterization of rotations
(as opposed to the integral of the density over the space being set equal to unity).
A very useful visual aid for this chapter is a Tinkertoy set! This allows you to
build a ball and stick model for the reference axes and for the crystal axes and move one
with respect to the other.
The major difference from a mathematical perspective between orientation and
misorientation lies in the symmetries that must be applied to them. A misorientation
always has to take account of the crystal symmetry on either side of the boundary. An
orientation has to take account of the crystal symmetry at one “end” of the rotation, and
the sample symmetry at the other “end” of the rotation. This can be expressed in the
following pair of equations, where S is a symmetry operator, and g is a rotation denoting
either orientation or misorientation. In these expressions, we have introduced a new
notation, ← ⎯ = → , which signifies “is physically indistinguishable from” in order to make
the point that two quantities involving symmetry operators may produce the results that
cannot distinguished from one another in the physical world (though they are not
mathematically identical). The first g is an orientation, and the second is a
misorientation, often written as ∆g, where the “∆” denotes difference in orientation.
g ← ⎯ = → S sample gS crystal
crystal crystal
g ← ⎯ = → SB gSA
The crystal symmetry is fixed by the packing of the atoms that comprise the material and
thus determine the crystal lattice. We shall use only point group symmetry and, in fact,
only the symmetry operators that are proper rotations (with determinant equal to +1).
Note however that distortions of the lattice (e.g. elastic distortions) or the presence of
defects can lower the actual crystal symmetry. Also, the sample symmetry is a
statistically based symmetry and applies only in an average sense to a polycrystal.
8/27/09 2