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11: Crystallography
11.5: Symmetry of Three Dimensional Atomic Arrangements

11.5: Symmetry of Three Dimensional Atomic Arrangements

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In the preceding sections, we discussed the shapes and symmetries of crystals. We now turn our attention briefly to space symmetry, the symmetry of three-dimensional atomic structures. What possible symmetry can be present? An atomic arrangement in a crystal consists of groups of atoms (an atomic motif) that repeat an infinite number of times according to one of 14 space lattices. Thus, the overall symmetry depends on both the arrangement of atoms in a motif and the lattice type.

Atomic motif symmetry may involve rotation axes, rotoinversion axes, mirror planes or inversion centers. So motifs may have symmetries equivalent to any of the 32 point groups. Consider an orthorhombic mineral, for example. Its atomic motif may have symmetry mmm, ²/_m²/_m²/_m, or mm2. Its lattice may be 222P, 222C, 222I, or 222F. That gives 12 possible combinations. If we do this calculation for all crystal systems, we come up with 61 possible symmetries for 3D atomic arrangements. But there are many more than that. Because in 3D, we find two additional symmetry operators that we have not considered previously.