¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

angles. For a completely orientation, the ranges of φ 1 and φ 2 are 0 through 2π, and the
range of Φ is π. Therefore when the expression in Eq. 3.N.1 above is integrated, the
volume is 8π 2 . Note that the choice of parameters dictates the form of the volume
element used to perform the integration. If we use degrees instead of radians, then the
normalization factor is 2*360 2 instead of 8π 2 . The volume element, or Haar measure,
changes depending on the parameters used.
3.N.2 Pole Figures projected from Orientation Distributions
Most of us encounter pole figures as the representation of experimental results obtained
on a special X-ray goniometer that permits the specimen to be rotated in two orthogonal
axes. A pole figure is a map of the intensity of diffraction from a particular set of
crystallographic planes as a function of declination/co-latitude (from the center of the
pole figure) and azimuth/longitude (sometimes labeled χ and φ, or Θ and φ). Having
developed the basis for an orientation distribution (OD), however, it is instructive to
consider how a pole figure can be constructed given a knowledge of the OD, i.e. a
mathematical definition. First we define a particular plane for which we would like to
construct a pole figure by the conventional Miller indices, (hkl). We then form a unit
vector, h, by normalizing the indices. Note that h represents a position on the unit sphere
so it is essentially spherical position information.
ˆ
h =
(h /a,k /b,l /c)
h 2 /a 2 + k 2 /b 2 + l 2 /c 2
We then need to relate the pole position to the corresponding position, w, in the pole
figure, which is € referred to specimen coordinates through the orientation of each grain
that contributes to (i.e. diffracts in) the pole figure. In this expression, R is a crystal
symmetry operator (unimodular matrix) and g-1 is the inverse (transpose) of the
orientation matrix described above.
€
w = g −1 ( ψ,Θ,φ )R (k) h
It is important to remember that this is a projection from a 3-parameter space into a 2parameter
space. This can be illustrated by reference to the standard mathematical
construction of a many-to-one mapping. Note that, although there are 24 proper rotation
operators in the cubic point group, the number of poles observed (on the upper
hemisphere) depends on the relationship of h to the symmetry elements ({001} coincide
with many of the rotation axes, e.g., and so only 3 distinct {001} poles are visible in the
upper hemisphere, unless one lies on the edge/equator); this effect is of course known as
multiplicity in crystallography. The relation is only simple for special choices of the
pole: the (001) pole, for example, eliminates the third Euler angle. In general to find the
range of values of Euler angles that correspond to a particular point in the pole figure (i.e.
fixing χ and φ) results in a relationship between the three angles. In fact a line in Euler
space is defined which is equivalent to a fiber.
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