¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

The resulting rotation can be converted to a matrix (passive rotation) by the following
expression, where δ is the Kronecker delta and ε is the permutation tensor.
a ij = δ ij cosθ + r i r j 1− cosθ
( ) + ε ijk r k sinθ
k =1,3
Note that the active rotation is obtained by changing the sign of the third term on the
RHS in order to obtain the transpose. The axis-angle pair can be recovered from a matrix
by finding the eigenvector associated with the only real eigenvalue of the matrix, and
examination of the trace of the matrix. The quantity norm is equal to 2sinθ. Note that
from a numerical perspective, the cosine formula for the angle is more accurate at large
angles, whereas the sine formula is more accurate for small rotation angles. Also, when
working with the inverse cosine function in computer programs, it is good practice to
check the magnitude of the argument and force it to lie between -1 and +1.
3.E Rodrigues vectors
norm = [ a(2,3) − a(3,2) ] 2 + [ a(1,3) − a(3,1) ] 2 + [ a(1,2) − a(2,1) ] 2
( r1 ,r2 ,r3 ) = ( a(2,3) − a(3,2),a(1,3) − a(3,1),a(1,2) − a(2,1) )/ norm
θ = cos −1 tr[ a]
−1
2
Another choice in more general use than Euler angles is that of Rodrigues vectors,
as popularized by Frank [Frank, F. (1988). “Orientation mapping.” Metallurgical
Transactions 19A: 403-408.], hence the term Rodrigues-Frank space for the set of
vectors. It has turned out to be quite useful for representation of misorientations for the
reason that many of the boundary types that correspond to a high fraction of coincident
lattice sites (i.e. low sigma values in the CSL model) occur on the edges of the Rodrigues
space and their coefficients are reciprocals of integer values. Some authors [Neumann, P.
(1991). “Representation of orientations of symmetrical objects by Rodrigues vectors”,
Textures and Microstructures 14-18 53-58] have pointed out that it has advantages for
texture representation also. Any set of points related by a common axis lie on a straight
line in Rodrigues space, for example, which has advantages for numerical manipulation
of orientation data. As a consequence of its use in misorientation representation, the
rotation matrix is denoted by ∆a, where the ∆ denotes a difference in orientation, i.e.
misorientation.
This representation can be thought of as an axis-angle representation in which the
direction of the R-F vector is parallel to the rotation axis and the magnitude of the vector
is the tangent † of the semi-angle (θ/2). The convenient features of R-F space include the
† Note that this scaling of the vector corresponds to the stereographic projection used in pole
figure representation.
∑
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