¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

more obvious that each column provides the equivalent in terms of expressing a sample
direction with coefficients based on the crystal frame of reference.
It is very important to remember that this definition of an orientation should be
interpreted as a transformation of axes (passive rotation) that converts tensor quantities
from the sample reference frame to the crystal frame. This means, for example, that
quantities known in the crystal frame, such as anisotropic elastic moduli, must be
converted to the sample frame with the inverse transformation. For the matrix
description provided above, the inverse is simply the transpose of the orientation matrix.
Be aware that no simple method exists of combining rotations when expressed in Miller
index form. Moreover, Miller index notation does not provide a continuous description
of rotations and is therefore not suitable for numerical descriptions of orientation.
Similarly axis-angle notation can provide a continuous description if the axis is described
as a unit vector with three real numbers, but not if the axis is described in terms of integer
Miller indices (with a finite number of digits).
3.B.2 Number of variables needed for an orientation.
We only need 3 variables to describe an orientation. This is hard to see in a 3x3
matrix with nine entries, though counting up the various constraints on the coefficients of
the matrix does provide the answer (orthogonality gives one set of 3 in terms of the other
6, i.e. reduces the number from 9 to 6; then each set of three direction cosines is
normalized, which reduces the number from 6 to 3). In fact, a matrix that describes a
rotation is termed an orthogonal matrix. It is much easier, however, to see the
requirement in terms of rotation angles and, specifically, Euler angles. Later on we will
show how axis-angle pairs, Rodrigues vectors, and quaternions are related to these
descriptions. An orthogonal matrix, a, is one for which one can write the following set of
six equations:
3.C Euler angles
2
2 ∑ aij =1, aij i
j
∑ =1 (3B.2.1)
Euler angles are the € most commonly used basis for representation of textures. They were
adopted early on in the development of texture analysis and it is convenient to develop a
familiarity with the locations of texture components of special interest for interpreting
experimental and simulation results. The reason for adopting them was that there was
already a large body of work on the physics of electrons and atoms that uses generalized
spherical harmonics. These (orthonormal) functions are most conveniently expressed in
terms of Euler angles. One very confusing point, however, is that different communities
have adopted different conventions for the definition of Euler angles (Bunge, Roe,
Canova, Kocks, etc.).
8/27/09 6