¶ 3. Mathematical Representation of Crystal Orientation, Misorientation

3.J Symmetry Operators (Matrices)
The standard way to approach symmetry operators is to represent them as matrices. First,
however, let’s recall some elementary ideas about symmetry operators from
crystallography. Rotational symmetry elements (which we will confine our attention to
here) exist whenever you can rotate a physical object and result is indistinguishable from
what you started out with. The following diagram is copied from Lecture Notes for 27-
201, Structures of Materials by Prof. M. De Graef (1996), fig. 9.2.
These rotations can be expressed in a simple mathematical form as unimodular matrices.
Except for the 3-fold axes in the trigonal and hexagonal groups, the effect of any of the 2fold
or 4-fold axes (or the 3-fold axes in the cubic point groups) is to permute the axes in
a simple fashion. 2-fold axes are called diads, 3-fold axes are triads and 4-fold axes are
quads. We will summarize the available operators in the table reproduced from the
Kocks/Wenk/Tomé book, Ch. 1, Table II.
Note how the essential features of the cubic point groups are based on the four 3-fold
axes on with the three 2-fold axes. Adding the 4-fold axes generates the more
highly symmetric cubic groups. Note also that the most symmetric group shown is 432,
rather than m3m, because the mirror elements have been omitted. The reasons for this
area discussed below. The symmetry elements have names and there are different names
for different contexts, as usual. In the context of crystallography, the operators are called
n-fold axes (or just n on diagrams) following the International notation. An alternative
notation is that of Schoenflies in which the operators are denoted by Cn, where n denotes
the order. This same notation occurs in discussions of group theory and quantum
n
mechanics. Yet another notation uses Luvw where uvw denotes the axis along which the
symmetry element operates, and n denotes the order of the axis.
For the purposes of discussing texture and misorientation, it is also important to
distinguish about which axis the rotation is performed. Thus a 2-fold axis about the z-
2
axis is known as a z-diad, or C2z, or L001 , a triad about [111] as a 111-triad etc.
Lastly, we need to mention the existence of symmetry operators of the second kind.
These operators include the inversion center and mirrors. The inversion simply reverses
any vector so that (x,y,z)->(-x,-y,-z). Mirrors operate through a mirror axis. Thus an xmirror
is a mirror in the plane x=0 and has the effect (x,y,z)->(-x,y,z). In matrix form,
the x-mirror is given by:
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