¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

"
2
h3 + (1 ! h3 )h2 $
th =
$
!(1! h3 )h1 h2 $
h1 1 ! h
2
#
3
2
!(1! h3 )h1h2 !h1 1! h %
3
2
2 '
h3 + (1! h3 )h1 h2 1 ! h3 '
!h2 1 ! h
2
3 h
'
3 &
Once the pole has been rotated onto the North pole, the crystal can then be rotated freely
about the pole without affecting the position of the pole. This rotation matrix is similar to
the intermediate rotation step described above for constructing the orientation matrix
from the Euler angles.
# cos! "sin! 0&
t ! = %
sin!
$ 0
cos!
0
0 (
1'
This is the key step in this development because this provides the free parameter over
which we can integrate in order to find the intensity in the pole figure, based on a line of
intensity in orientation space. A rotation, tf, about the North pole is simply tf =
tf([001],ζ), where ζ is the (free) rotation angle. Finally, we need to rotate the pole onto
the position in the pole figure that is of interest. This is accomplished by successive
rotations about [100] by χ and then [001] by φ.
# cos! "sin! 0&
t ! = %
sin!
$ 0
cos!
0
0 (
1'
,t ) =
# 1 0 0 &
%
0
$ 0
cos )
sin )
"sin ) (
cos) '
The product of this set of rotations must then be equal to the orientation matrix, i.e.
g(φ1,Φ,φ2) = tφ•tχ•tζ•th = t(h,ζ,w)
where the matrix on the RHS contains only one free parameter, i.e. the rotation by ζ
about the pole h. Alternatively we can write the following for the axis transformation.
a(φ1,Φ,φ2) = (tφ•tχ•tζ•th) T = t T (h,ζ,w)
This then allows us to write the following for the intensity, P, at w(χ,φ) in the pole figure.
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