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The 14 3D Bravais Lattices

This Demonstration shows the characteristics of 3D Bravais lattices arranged according to seven crystal systems: cubic, tetragonal, orthorhombic, monoclinic, triclinic, rhombohedral and hexagonal. Each crystal system can be further associated with between one and four lattices by adding to the primitive cell (click "P"): a point in the center of the cell volume (click "I"), a point at the center of each face (click "F") or a point just at the center of the base faces (click "C"). The points located at the center/faces are highlighted in blue; each point is also a vertex or center of the cell/face, therefore each point is equivalent to every other point.
Crystal systems are determined by the relative lengths of the basis vectors , , and the angles between them [1].
It is possible to shift the cell by one unit along a basis vector by selecting the , , values. When repeated, this can generate the entire lattice.

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The Bravais lattice theory establishes that crystal structures can be generated starting from a primitive cell and translating along integer multiples of its basis vectors, in all directions.
Snapshot 1: This shows the primitive cubic system consisting of one lattice point at each corner of the cube. Each atom at a lattice point is then shared equally between eight adjacent cubes, and the unit cell therefore contains in total one atom . The crystal structure of pyrite is primitive cubic [2].
Snapshot 2: The face-centered cubic system has lattice points on the faces of the cube; each contributes exactly , in addition to the corner lattice points, giving a total of four lattice points per unit cell ( from the corners plus from the faces). The crystal structure of sodium chloride is face-centered cubic [2].
Snapshot 3: This shows the hexagonal system. Graphite is an example of the hexagonal crystal system [2].
References
[1] M. de Graef and M. E. McHenry, Structure of Materials: An Introduction to Crystallography, Diffraction and Symmetry, Cambridge: Cambridge University Press, 2007.
[2] W. B. Pearson, A Handbook of Lattice Spacings and Structures of Metals and Alloys, New York: Pergamon Press, 1958.
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