¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

Let’s take the example of the 2-fold symmetry on the x-axis which is best understood as
the limiting case of θ→∞ for [1,0,0]tanθ/2. If we consider a point lying on the symmetry
plane at x=-1, i.e. ρ A = [-1,ρ 2,ρ 3], then the action of the symmetry operator is to take it to
the following new point, ρ C:
ρC = −1,ρ 2 ,ρ [ 3]
+ tanθ / 2[ 1,0, 0]
− tan θ / 2[0, ρ3 ,−ρ 2 ]
1+ tanθ / 2
As we take this to the limit of θ→∞ then ρ C tends towards the point [+1,ρ 3,-ρ 2] which is
the original point reflected through x=0 but also turned through 90°! By considering a
point that is just inside the “back plane” (x=-1), i.e. ρ A = [-1+ε,ρ 2,ρ 3], then evaluation of
the same combination shows that the related point lies just outside the front plane, x=+1,
and vice versa. Note that the planes associated with the symmetry elements are not
mirror planes, as we are accustomed to seeing them in stereographic projections of point
groups, for example.
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