Space Group

The point symmetry operations - mirrors, rotations, and inversion - combine to produce the 32 point groups, which are sufficient to describe everything observable about the distribution of faces around the exterior of a crystal. ^[1] But crystal faces are only the surface expression of a structure that repeats continuously in three dimensions, and capturing that internal repetition requires a second type of symmetry: translation. Simple translation in three dimensions produces the 14 Bravais lattices. ^[1] The 230 space groups arise from combining these two: they represent every possible way of repeating point group symmetry through a three-dimensional lattice. ^[1] Every mineral belongs to exactly one of these 230 space groups. ^[1]

Combining Point Groups with Lattices

The construction of space groups begins by placing a group of atoms - one that already possesses the symmetry of one of the 32 point groups - at every node of a compatible Bravais lattice. ^[1] The word “compatible” matters here: the symmetry of the point group must match the symmetry of the lattice. An isometric point group can only be placed on an isometric lattice; a tetragonal point group on a tetragonal lattice; and so on for each crystal system. ^[1] Combining each of the 32 point groups with the lattices it is compatible with - without any additional symmetry operations - produces 73 of the 230 space groups. ^[1]

The remaining 157 space groups require two additional symmetry operations that only become possible in three dimensions: glide planes and screw axes. ^[1] Both are compound operations - each combines a translation with either reflection or rotation. ^[1] Neither can be observed from a crystal’s external faces, which is precisely why point symmetry alone cannot fully describe the internal structure.

Glide Planes

A glide plane is the result of translating a structural unit by a fixed distance and direction, then reflecting it across a plane called the glide plane. ^[1] The two steps - translation then reflection - happen together as a single operation. Neither step alone would reproduce the pattern; only the compound action does.

The pattern left by footprints is a good physical analogy. Each footprint follows the previous one at a regular spacing, but successive prints alternate between left and right feet - and left and right feet are mirror images of each other. ^[1] The footprint pattern could not be generated by pure translation (which would require the same foot each time) or pure reflection (which would not advance the position). The glide is the only operation that produces it.

In minerals, glide planes operate at the atomic scale. The silicon-oxygen chains that run through pyroxene crystals are a direct example: each silicon tetrahedron in the chain is related to the next by a translation followed by a reflection, so the tetrahedra alternate orientation along the chain length. ^[1] Glide planes in a mineral must always be parallel to the mirror planes that appear in the mineral’s point group, because the point symmetry constrains where compound operations can occur. ^[1]

Screw Axes

A screw axis combines translation with rotation - a structural unit is shifted by a fixed distance and simultaneously rotated, repeating the pattern in a helical path through the crystal, like the treads of a spiral staircase. ^[1] The allowed rotation angles are the same as for ordinary rotation axes: 2-fold (180°), 3-fold (120°), 4-fold (90°), and 6-fold (60°). ^[1]

A 4-fold screw axis repeats a 90° wedge of the crystal structure at every 90° step, advancing in both angle and position simultaneously rather than rotating in place. ^[1] The remaining symmetry operations of the mineral then fill in the full structure around this helical core.

Quartz provides the most familiar mineral example. Its silicon tetrahedra are stacked along the c axis by a 3-fold screw: each tetrahedron is displaced by a fixed distance along c and rotated 120°, building a continuous helix of tetrahedra that winds through the crystal. ^[1] This helix can wind either clockwise or counterclockwise, which is why quartz occurs in left-handed and right-handed forms - the two are mirror images of each other at the atomic level. Screw axes must always be parallel to the rotation axes already present in the mineral’s point group, since the compound operation cannot conflict with the simpler rotational symmetry it extends. ^[1]

References

1. Nesse, W. D. (2018). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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