Name | Number of Bravais lattices | Conditions |
Triclinic | 1 |
a_{1} ¹
a_{2} ¹
a_{3}
a ¹ b ¹ g |
Monoclinic | 2 |
a_{1} ¹
a_{2} ¹
a_{3}
a = b = 90° ¹ g |
Orthorhombic | 4 |
a_{1} ¹
a_{2} ¹
a_{3}
a = b = g = 90° |
Tetragonal | 2 |
a_{1} =
a_{2} ¹
a_{3}
a = b = g = 90° |
Cubic | 3 |
a_{1} =
a_{2} =
a_{3}
a = b = g = 90° |
Trigonal | 1 |
a_{1} =
a_{2} =
a_{3}
a = b = g < 120° ¹ 90° |
Hexagonal | 1 |
a_{1} =
a_{2} ¹
a_{3}
a = b = 90° g = 120° |
Lattice type | Number of lattice points/atoms per unit cell |
Nearest distance between lattice points |
Maximum packing density | Example |
Simple cubic | 1/1 | a | p/6 = 52 % | Phosphor |
Body centered cubic | 2/2 | aÖ3/2 | pÖ3/8 = 68 % | Tungsten |
Face centered cubic | 4/4 | aÖ2/2 | pÖ2/3 = 74 % | Aluminum |
Diamond | 4/8 | aÖ2/2 Nearest distance between atoms: aÖ3/4 |
pÖ3/16 = 34 % | Silicon |
Cubic lattices have the highest degree of symmetry of any Bravais lattice. They belong to the (m3m) symmetry group which contains the following symmetry groups and operations:
Identity | 1 | |
Three equivalent axis of two-fold rotation | 3[2^{|}] | [100], [010], [001] |
Six equivalent axis of four-fold rotation | 6[4^{|}] | [100], [010, [001], [-100], [0-10], [00-1] |
Six equivalent axis of two-fold rotation | 6[2] | [110], [101], [011], [1-10], [10-1], [01-1] |
Eight equivalent axis of three-fold rotation | 8[3] | [111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [-11-1], [1-1-1] |
Inversion | -1 | |
Three equivalent mirror planes | 3[m^{|}] | [100], [010], [001] |
Six equivalent axis of four-fold rotation with inversion | 6[-4] | [100], [010, [001], [-100], [0-10], [00-1] |
Six equivalent mirror planes | 6[m] | [110], [101], [011], [1-10], [10-1], [01-1] |
Eight equivalent axis of three-fold rotation with inversion | 8[-3] | [111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [-11-1], [1-1-1] |
Note that the (m3m) symmetry group is the highest possible symmetry group associated with a cubic crystal. A limited symmetry of the basis (the arrangment of atoms associated with each lattice point) can yield a lower overall symmetry group of the crystal.