¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

(ρ1, ρ2) = {ρ1 + ρ2 - ρ1 x ρ2}/{1 - ρ1•ρ2} (3.L.14.i)
If one of the vectors defines a line (through the origin) by multiplying a scalar, e, by a
unit vector, m, then the product takes the form,
(ρ1, em) = ρ1 + e/{1 - eρ1•m}[ρ1(ρ1•m) + m - ρ1 x m}] (3.L.14.ii)
The term in the square brackets is a vector that does not depend on the scalar, e. The
term in the curly brackets is a scalar that does depend on e. Thus the rotation of the
straight line has produced another straight line. By extension, planes contain straight
lines and so planes also rotate into planes.
ρ A ’
3.L.14b Constructing the bisecting orientation between two orientations
To construct the bisecting orientation, ρbisect, we follow the development given by S&B
on p11. First form the misorientation between A and B, i.e. the rotation that carries one
from B to A.
0
∆g BA = (ρA,- ρB)
The rotation angle associated with this misorientation is defined as θ dis, and the rotation
axis is defined as Δ ˆ
g BA , given by:
tan(θdis/2) = |∆gBA| Δg ˆ BA = ∆gBA/|∆gBA| Then transform the misorientation by half of its rotation angle (in order to obtain equal
rotation angles between each orientation and the new, bisecting orientation) to obtain a
new rotation, ∆ρAB, where the “∆” merely signifies that we will use it to construct the
new orientation.
∆ρAB = tan(θ dis/4) Δ ˆ
g BA (3.L.14b.i)
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