Bravais Lattice

When the five two-dimensional plane lattices are extended into three dimensions by adding a third translation vector, the result is 14 distinct three-dimensional space lattices, known as the Bravais lattices. ^{[1] [1]} These 14 lattices represent every geometrically distinct way that points can be arranged in three-dimensional space with translational periodicity. No other arrangements are possible - any crystal, however complex, uses one of these 14 lattice types.

The 14 Lattices and Six Crystal Systems

The 14 possible translational arrays separate into six distinct geometric categories according to unit-cell symmetry. ^[1] This half-dozen set of underlying architectural blocks forms the basis of the crystal systems we observe: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and isometric (cubic). ^[1] The unit cell shapes are the shapes that can be regularly stacked together without leaving voids - familiar forms like cubes, rectangular boxes, and prisms. ^[1]

Within the 14 lattices, four types of unit cell are distinguished by where lattice nodes are located: ^[1]

• Primitive (P): nodes at corners only ^[1]
• Body-centered (I): corners plus one node at the centre ^[1]
• Face-centered (C): corners plus nodes on two opposite faces ^[1]
• All face-centered (F): corners plus nodes at the centre of every face ^[1]

The distinction between primitive and centred lattices matters because the extra lattice nodes change the symmetry relationships within the cell, which affects how X-ray diffraction data are interpreted. ^[1]

Identifying Lattices

In a hand sample it is not possible to distinguish between the different Bravais lattices within the same crystal system - that requires X-ray diffraction. ^[1] What can be determined from a hand sample is the crystal system, based on the symmetry of the crystal faces. Knowing the crystal system narrows the possible Bravais lattices to the two or three that belong to that system, but the specific lattice type within the system can only be resolved instrumentally. This is why the Bravais lattice is essential background knowledge for X-ray crystallography but is not a property you can read directly from a specimen.

References

1. Nesse, W. D. (2017). Introduction to Mineralogy, 3rd ed. Oxford University Press.

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