Page 71 - Materials Chemistry, Second Edition

58 2 Solid-State Chemistry

Figure 2.35. Illustrations of: (a) 6, (b) 4, and (c) m crystallographic symmetry operations.

by reflection through a center of symmetry. If individual lattice points are simply
reflected through a plane of symmetry, the operation is symbolized by m, denoting
the presence of a mirror plane. Although individual atomic and ionic lattice points
are not affected by n and m operations, molecular lattice points exhibit a change in
handedness following these operations (see Figure 2.35b, c). Point groups that
include these operations, known as improper symmetry operations, must exclude
all chiral molecules, as they would then be superimposible on their mirror images.
Two important restrictions apply to crystallographic point group symmetry
operations:
1. The symmetry operations must be compatible with infinite translational repeats
in a crystal lattice;
2. A symmetry operation cannot induce a higher symmetry than the unit cell
possesses.
The point group symmetry describes the non-translational symmetry of the crystal;
however, the infinite crystal lattice is generated by translational symmetry (see
below). Only two, three, four and sixfold rotation axes are compatible
with translational symmetry, so point groups containing other types of rotation