¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

3.B.1 Miller index representation of orientation.
Miller indices can be used to represent an orientation and this is in fact the
standard method in the materials literature to describe particular (“ideal”) orientations
(“texture components”). The representation takes two forms, one for a full description of
orientation (sometimes referred to as “bi-axial alignment” in the thin film literature), and
the second for an implied fiber texture. In the first form, a sample is assumed to have a
rectangular shape with large enough aspect ratio that a place normal can be associated
with the face with largest area, and a direction with the longest dimension. In the case of
rolled sheets, these two are the rolling plane normal (ND, or RPN) and the rolling
direction (RD), respectively.
To describe an orientation, the convention is to specify the crystallographic plane
normal that is parallel to the specimen normal (e.g. the ND) and a crystallographic
direction that is parallel to the long direction (e.g. the RD). Looking ahead, this is akin to
working in the space of an inverse pole figure, in that we specify which crystal direction
can be found when we look along a fixed sample direction.
(hkl) // ND, [uvw] // RD, or (hkl)[uvw]
In effect, one is writing down the direction cosines † of the specimen’s third and first axes
with respect to the crystallographic axes. Think about this carefully because, although
one has used a crystallographic description, the quantities described are associated with
the reference frame fixed in the specimen.
† See Ch. 2 for definitions of quantities such as direction cosines
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