¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

Analysis. He labels the crystal axes as K A and the crystal axes as K B; the specimen axes
are X, Y & Z, and the crystal axes are X’, Y’ & Z’.
3.C.2 Correspondence between Matrices
Given the orientation matrix derived from Euler angles and the matrix derived from
direction cosines, one can immediately see how to convert between the various
descriptions because each pair of corresponding entries must be identical to each other.
That is to say, b 1 in the first matrix = cosφ 1 cosφ 2− sinφ 1 sinφ 2 cosΦ, etc.
b1 Sample
t1 n1 ⎛ ⎞
⎜ ⎟
aij = Crystal
⎜
b2 t2 n2 ⎟
⎜ ⎟
⎝ b3 t3 n3 ⎠
⎛ cosϕ1 cosϕ 2 − sinϕ1 sinϕ 2 cosΦ sinϕ1 cosϕ 2 + cosϕ1 sinϕ 2 cosΦ
⎜
≡
⎜
−cosϕ1 sinϕ 2 − sinϕ1 cosϕ 2 cosΦ −sinϕ1 sinϕ 2 + cosϕ1 cosϕ 2 cosΦ
⎜
€
⎝ sinϕ1 sinΦ −cosϕ1 sinΦ
sinϕ 2 sinΦ⎞
⎟
cosϕ 2 sinΦ
⎟
cosΦ ⎠
This permit Euler angles to be converted to Miller indices by extracting the first and third
€ columns of the orientation matrix, and re-scaling each (unit) vector to have suitable
integer values.
More conversions are given below.
3.C.3 Other definitions € of Euler angles
h = n sin Φ sinϕ 2
k = n sin Φ cosϕ 2
l = n cosΦ
( )
u = n ′ cosϕ1 cosϕ 2 − sinϕ1 sinϕ 2 cosΦ
v = n ′ ( − cosϕ1 sinϕ 2 − sinϕ1 cosϕ 2 cosΦ)
w = n ′ sin Φ sinϕ1 The other conventions are those of Roe [Roe, R. J. (1965). “Description of
Crystallite Orientation in Polycrystalline Materials. 3. General Solution to Pole Figure
Inversion”, Journal of Applied Physics 36 2024], who developed an analysis in parallel
with Bunge), Matthies and Kocks. The main difference between the Bunge convention
and the others is that the second rotation (Θ above) is about the (rotated) X-axis, whereas
in all the other conventions it is about the (rotated) Y-axis. From the (Russian) physics
literature, Borisenko and Tarapov (Vector and Tensor Analysis, Dover) call them the
precession, nutation and pure rotation angles, respectively, and follow the Bunge
convention. Other physics literature follows the Roe convention; it would be interesting
to know how the west and east came to take different approaches to Euler angles!
8/27/09 8