Rotation — A Fundamental Crystallographic Symmetry Operation
Rotation — A Fundamental Crystallographic Symmetry Operation
The geometric operation by which a crystal can be brought into self-coincidence by turning about an axis
Rotation, in crystallography, is the symmetry operation by which a crystal is turned about an axis through a defined angle and brought into self-coincidence — that is, into a configuration indistinguishable from the starting position. Rotation is one of the small number of geometric operations from which all crystallographic symmetry is built, and the rotation axes admitted by the translational symmetry of crystal lattices determine the partition of crystals into the seven crystal systems and the thirty-two crystal classes that gemmologists and mineralogists use to describe and identify crystalline solids.
Geometry
The angle through which a crystal must be rotated to reach self-coincidence is the inverse of the fold of the rotation axis. A rotation axis of fold n is one for which a rotation through 360°/n returns the crystal to a configuration indistinguishable from the original. The crystallographically admissible folds are 1, 2, 3, 4, and 6. A 1-fold axis (rotation through 360°, equivalent to no rotation) is the identity operation and is present in every object. A 2-fold axis requires a rotation of 180°; a 3-fold axis, 120°; a 4-fold axis, 90°; and a 6-fold axis, 60°.
Rotation axes of fold 5, 7, and higher than 6 are not compatible with the periodic translational symmetry of crystal lattices and therefore cannot occur in classical crystals. This restriction — known as the crystallographic restriction theorem — is the geometric reason that crystals fall into a finite, small number of symmetry classes rather than into a continuum. Quasicrystals, discovered in the early 1980s and the subject of the 2011 Nobel Prize in Chemistry, demonstrate that 5-fold symmetry can occur in solids without periodic translational lattice — a discovery that extended rather than overturned the classical theorem.
Combination with other operations
Rotation combines with two other elementary symmetry operations — inversion through a centre point and reflection across a plane — to generate the full set of crystallographic symmetry operations. A rotation followed by an inversion is termed a rotoinversion or improper rotation; a rotation followed by a reflection across the rotation plane is a rotoreflection. Together with translations, these operations generate the 230 space groups that describe all possible periodic crystal structures.
The rotation axes of a crystal can be combined: a cubic crystal possesses three mutually perpendicular 4-fold axes through the centres of opposite faces, four 3-fold axes through opposite corners, and six 2-fold axes through opposite edge midpoints, all simultaneously. The hexagonal system carries a single 6-fold axis along its principal direction. The trigonal system carries a 3-fold axis. Rotation axes of fold 2 alone — without higher-fold axes — characterise the orthorhombic, monoclinic, and triclinic systems in decreasing order of symmetry richness.
Relevance to gemmology
The rotation axes of a gem species determine its optical and physical anisotropy. Crystals with a single high-fold axis — quartz, beryl, corundum, tourmaline — are uniaxial in their optical character, with a unique optic axis aligned along the high-fold rotation axis. Crystals with multiple equivalent high-fold axes — cubic species such as diamond, garnet, spinel — are isotropic. Crystals with no rotation axis higher than 2-fold are biaxial. The pleochroism, refractive-index behaviour, and birefringence of gem materials all derive from the rotational symmetry of the underlying crystal structure, and the standard gemmological observations of optic figure, optic sign, and pleochroism are direct consequences of these symmetries.
For the gemmologist, rotation is therefore not an abstract concept but the underlying reason that quartz shows a uniaxial-positive interference figure, that diamond shows none, and that olivine shows a biaxial figure with characteristic 2V angle. The crystal-system classification used in every gemmological reference rests directly on the geometry of rotation.