Hexagonal lattice | Wallpaper group p6m | Unit cell |
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The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. 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,
| a 1 | = | a 2 | = a . {\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
g = 4 π a 3 . {\displaystyle g={\frac {4\pi }{a{\sqrt {3}}}}.}Honeycomb point set
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis.2 The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb point set.
Crystal classes
The hexagonal lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
Geometric class, point group | Wallpaper groups | ||||
---|---|---|---|---|---|
Schön. | Intl | Orb. | Cox. | ||
C3 | 3 | (33) | [3]+ | p3(333) | |
D3 | 3m | (*33) | [3] | p3m1(*333) | p31m(3*3) |
C6 | 6 | (66) | [6]+ | p6(632) | |
D6 | 6mm | (*66) | [6] | p6m(*632) |
See also
- Square lattice
- Hexagonal tiling
- Close-packing
- Centered hexagonal number
- Eisenstein integer
- Voronoi diagram
- Hermite constant
References
Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18. https://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf ↩
Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18. https://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf ↩