Hexagonal lattice

From Wikipedia, the free encyclopedia
Equilateral Triangle Lattice.svg Wallpaper group diagram p6m.svg 2d hp.svg
Hexagonal lattice Wallpaper group p6m Unit cell

The hexagonal lattice (sometimes called 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,

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

Honeycomb point set[edit]

Honeycomb point set as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors and are primitive translation vectors.

The honeycomb point set 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 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[edit]

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 Arithmetic
class
Wallpaper groups
Schön. Intl Orb. Cox.
C3 3 (33) [3]+ None p3
(333)
 
D3 3m (*33) [3] Between p3m1
(*333)
p31m
(3*3)
C6 6 (66) [6]+ None p6
(632)
 
D6 6mm (*66) [6] Both p6m
(*632)
 

See also[edit]

References[edit]

  1. ^ a b Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18.