11.6: Space Groups

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When we combine the space group operators in the tables above with the 14 possible space lattices, we get 230 possible space groups. See Box 11-2. They represent all possible symmetries crystal structures can have, although most of them are not represented by any known minerals. Deriving them all is not trivial, and crystallographers debated the exact number until the 1890s when several independent studies concluded that there could only be 230.

Crystallographers use several different notations for space groups; the least complicated is that used in the International Tables for X-ray Crystallography (ITX) (Hahn, 1983). ITX space group symbols consist of a letter indicating lattice type (P, I, F, R, A, B, or C) followed by symmetry notation similar to conventional Hermann-Mauguin symbols. An example is P4₂/m 2₁/n 2/m, the space group of rutile. Rutile has a primitive (P) tetragonal unit cell, a 4₂ screw axis perpendicular to a mirror plane, a 2₁ screw axis perpendicular to an n glide plane, and a proper 2-fold axis perpendicular to a mirror plane. Rutile crystals have symmetry ⁴/_m²/_m²/_m.

Or consider garnet. Garnet crystals have point group symmetry ⁴/_m3²/_m. Garnet’s space group is I4₁/a 3 2/d. This describes a body-centered unit cell, a 4₁ screw axis perpendicular to an a glide plane, a 3-fold rotoinversion axis, and a proper 2-fold axis perpendicular to a d glide plane. Rather than using an entire symbol, crystallographers often use abbreviations for space groups (and occasionally for point groups). Thus, they would say that garnet belonged to the space group Ia3d. This shorthand notation, however, can be confusing for those not familiar with ITX conventions.

The translations associated with lattices, glide planes, and screw axes are very small, on the order of tenths of a nanometer, equivalent to a few angstroms. Detecting their presence by visual examination of a crystal is impossible, with or without a microscope. A crystal with symmetry ⁴⁄_m²/_m²/_m could belong to the space group I4₁/a 2/c 2/d, but there are also 19 other possibilities. We say that the 20 possibilities are isogonal, meaning that when we ignore translation they all have the same symmetry. Without detailed X-ray studies, telling one isogonal space group from another is impossible, and we are left with only the 32 distinct point groups.

For some supplementary information and a different perspective on space groups, check out the video that is linked here:

Video 11-4: Space group operators and space groups (8 minutes)

Why Are There Only 230 Space Groups?

In Chapter 10 we showed that point groups may have one of 32 symmetries; in this chapter we determined that crystal structures must have one of 14 Bravais lattices. When we combine point groups with Bravais lattices, and consider all possible space group operators, we get 230 possible space groups. The 230 space groups are the only 3D symmetries that a crystal structure can have. They were tabulated in the 1890s by a Russian crystallographer, E. S. Federov; a German mathematician, Artur Schoenflies; and a British amateur, William Barlow, all working independently.

Why are there only 230 space groups? The answer is that symmetry operators, as we have already seen, can only combine in certain ways. In the discussion of point group symmetry, we concluded that mirrors and rotation axes can only combine in 32 ways. Some combinations required that other symmetry be present. Other combinations were redundant or led to infinite symmetry, which is impossible. When we discussed the 17 possible plane symmetry groups, we also concluded that symmety operators and plane lattices can only combine is a limited number of ways. The same is true of space group operators and Bravais lattices; only certain combinations are allowed and some combinations require other symmetry to be present.

Triclinic lattices (P1 and P1) may not be combined with 2-fold axes of any sort. Similarly, 3, 3₁, and 3₂ axes are only consistent with a rhombohedral or hexagonal lattice (3R or 6P). If an atomic arrangement contains two perpendicular 4-fold axes, it must contain a third and it must contain 3-fold axes, too. And, we just showed that the presence of a 4₂ axis require that a 2-fold axis is present. There are many other constraints, too, which is why the total number of space groups is only 230.

Although minerals may belong to any of the 230 possible space groups, they are not evenly distributed. In fact, there are no known minerals that fall into many of the groups. Chapter 14 of this book has descriptions of 180 of the most common minerals. They fall into only 71 different space groups. About a third of all minerals belong to one of the monoclinic space groups.