Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete symmetry group. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry.

Related Topics:
Space group - Symmetry - Translational symmetry - Symmetry group - Crystallography - Crystal - Euclidean geometry

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There are 7 crystal systems:

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• Triclinic, all cases not satisfying the requirements of any other system; thus there is no other symmetry than translational symmetry, or the only extra kind is inversion.
• Monoclinic, requires either 1 two-fold axis of rotation or 1 mirror plane.
• Orthorhombic, requires either 3 two-fold axes of rotation or 1 two fold axis of rotation and two mirror planes.
• Tetragonal, requires 1 four-fold axis of rotation.
• Rhombohedral, also called trigonal, requires 1 three-fold axis of rotation.
• Hexagonal, requires 1 six-fold axis of rotation.
• Isometric or cubic, requires 4 three-fold axes of rotation.

There are 2, 13, 59, 68, 25, 27, and 36 space groups per crystal system, respectively, together 230.

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Within a crystal system there are two ways of categorizing space groups:

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• by the linear parts of symmetries, i.e. by crystal class, also called crystallographic point group; each of the 32 crystal classes applies for one of the 7 crystal systems
• by the symmetries in the translation lattice, i.e. by Bravais lattice; each of the 14 Bravais lattices applies for one of the 7 crystal systems.

The 73 symmorphic space groups (see space group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.

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Crystal system: Crystallographic point group ►