Solid State Theory

lattice + basis

A(r) = A(r+T) with T = n1*a1 + n2*a2 + n3*a3 n1,2,3 are integer

• The vectors are NOT unique.
• There are 5 Bravais lattices in 2D and 14 in 3D 

A primitive cell fills the whole space without holes or overlapping.

• Parallelogram/-epipid based on the Bravais vectors. (not unique)
• every other shape which fills the whole space without holes or overlapping.
• Wigner-Seitz cell (rectangular or hexagon)

All primitive cells have the same Volume and have one lattice point. (Proof it)
(There are also non primitive cells which may have higher symmetry, body centered cube bcc)

is no Bravais lattice.

Graphene has a honeycomb lattice (two hexagonal lattices )It can be discribed as a hexagonal  lattice with a basis of 2 atoms.

There is not only Translational symmetrie but also rotational.

There are only 2, 3,4,6 fold rotation symmetries allowed (Proof it).

(5 is possible in quasi crystals)

• oblique (least symmetry) phi != 90 |a1| != |a2|
• rectangular phi = 90 |a1| != |a2| ( reflection and 180 Rotation)
• square phi =90 |a1| = |a2| (90 Rotation)
• hexagonal phi =60 |a1| = |a2| (60 Rotations)
• centered rectangular rhombic