Page 74 - Materials Chemistry, Second Edition

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2.3. The Crystalline State

mirror planes that are parallel to the rotation axis. In contrast the 6/mmm point group
has a mirror plane perpendicular to the rotation axis (in the plane of the paper), as
well as other mirror planes that lie parallel to the primary rotation axis. Carrying out
the first two symmetry operations automatically generates the third term of the point
group symbol, if present. For instance, for the mm2 point group, the twofold rotation
axis is generated by the presence of two mutually perpendicular mirror planes.
The 32 crystallographic point groups are useful to define the contents of a discrete
unit cell; however, there must be additional symmetry elements that take into
account the translational symmetry of an extended periodic lattice. Accordingly,
both glide planes and screw axes feature translation in addition to mirror and
rotation operations, respectively (Figure 2.38). For glide planes, the translation is
designated as a, b,or c if movement is halfway along the a, b,or c unit cell axes,
respectively. If the translation is along the diagonals 1=2(a þ b), 1=2(a þ c), or
1=2(b þ c), the glide plane is designated as n. Lastly, if the translation is along 1=4
(a þ b), 1=4(a þ c), or 1=4(b þ c), the glide plane is given the symbol d. [37]

Figure 2.38. Illustrations of: (a) 2 1 screw axis (twofold rotation axis and 1/2 translation), listing the
orientations and (x, y, z) coordinates for the original/translated object, (b) glide plane.