Bravais Lattices and Crystal Systems

Every periodic crystal is built on one of just fourteen distinct space lattices, grouped into seven crystal systems by their symmetry, which together exhaust the ways points can fill space periodically.

Definition

A Bravais lattice is an infinite array of discrete points generated by integer combinations of primitive translation vectors, looking identical from every point; the fourteen distinct Bravais lattices, organized into seven crystal systems by symmetry, exhaust all periodic point arrangements in three dimensions.

Scope

This topic develops the Bravais lattice as the set of all translation vectors leaving a crystal invariant, the choice of primitive and conventional unit cells, and the classification of lattices into seven crystal systems and fourteen Bravais types. It covers point-group symmetry, the distinction between primitive, body-centered, face-centered, and base-centered cells, and the role of the basis in completing a crystal structure. It excludes the reciprocal-space and diffraction treatment handled in sibling topics.

Core questions

• What property defines a Bravais lattice, and why are there exactly fourteen of them?
• How do the seven crystal systems group the lattices by their point-group symmetry?
• When is a conventional (centered) cell preferred over the smaller primitive cell?
• How does adding a basis to a Bravais lattice produce a real crystal structure?

Key concepts

• Primitive translation vectors and unit cells
• Fourteen Bravais lattices
• Seven crystal systems
• Centered cells: body-, face-, and base-centered
• Lattice plus basis as a crystal structure

Clinical relevance

The Bravais classification is the indexing scheme for all crystalline materials; it organizes the structures of metals, minerals, and compounds and is the starting point for every calculation of diffraction patterns, electronic bands, and lattice vibrations.

History

Frankenheim proposed fifteen lattice types in 1842; Bravais corrected the count to fourteen in 1850 by recognizing that two of them were equivalent, giving the classification of space lattices that still carries his name.

Key figures

• Auguste Bravais
• Moritz Frankenheim

Seminal works

• ashcroft1976
• kittel2005

Frequently asked questions

Why are there only fourteen Bravais lattices?

The requirement that the lattice be both periodic and compatible with one of the allowed crystallographic point symmetries restricts the possibilities; enumerating the distinct combinations of system and centering yields exactly fourteen non-equivalent space lattices.

Is the unit cell unique?

No. Many cells can generate the same lattice; the primitive cell contains exactly one lattice point and has the smallest volume, while conventional centered cells are often chosen because they display the full symmetry more clearly.