Hexagonal lattice

Two-dimensional Bravais lattices:
1 – oblique (monoclinic),
2 – rectangular (orthorhombic),
3 – centered rectangular (orthorhombic),
4 – hexagonal,
5 – square (tetragonal).

The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types.^[1] The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,

${\displaystyle |\mathbf {a} _{1}|=|\mathbf {a} _{2}|=a.}$

The reciprocal lattice of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length

${\displaystyle g={\frac {4\pi }{a{\sqrt {3}}}}.}$

Honeycomb lattice[]

Honeycomb lattice as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors ${\displaystyle \mathbf {a} _{1}}$ and ${\displaystyle \mathbf {a} _{2}}$ are primitive translation vectors.

The honeycomb lattice is a special case of the hexagonal lattice with a two-atom basis.^[1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb lattice can be seen as the union of two offset triangular lattices.

In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb lattice.

See also[]

• Square lattice
• Hexagonal tiling
• Close-packing
• Centered hexagonal number
• Eisenstein integer
• Voronoi diagram

References[]

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