**a**,

_{1}**a**and

_{2}**a**and a set of integers k, l and m so that each lattice point, identified by a vector

_{3}**r**, can be obtained from:

In two dimensions there are five distinct Bravais lattices, while in three dimensions there are fourteen. These fourteen lattices are further classified as shown in the table below where a

_{1}, a

_{2}and a

_{3}are the magnitudes of the unit vectors and a, b and g are the angles between the unit vectors.

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° |

## Cubic lattices

Cubic lattices are of interest since a large number of materials have a cubic lattice. There are only three cubic Bravais lattices. All other cubic crystal structures (for instance the diamond lattice) can be formed by adding an appropriate base at each lattice point to one of those three lattices. The three cubic Bravais lattices are the simple cubic lattice, the body centered cubic lattice and the face centered cubic lattice. A summary of some properties of cubic latices is listed in the table below:Lattice type | Number oflattice points/atoms per unit cell | Nearest distancebetween 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 |

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] |

### Simple cubic lattice

The simple cubic lattice consists of the lattice points identified by the corners of closely packed cubes.sc.gif

### Body centered cubic lattice

The body centered lattice equals the simple cubic lattice with the addition of a lattice point in the center of each cube.bcc.gif

### Face centered cubic lattice

The face centered lattice equals the simple cubic lattice with the addition of a lattice point in the center of each of the six faces of each cube.fcc.gif

### Diamond lattice

The diamond lattice consist of a face centered cubic Bravais point lattice which contains two identical atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of diamond, germanium and silicon.diamond.gif

### Zincblende lattice

The zincblende lattice consist of a face centered cubic Bravais point lattice which contains two different atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of zincblende (ZnS), gallium arsenide, indium phosphide, cubic silicon carbide and cubic gallium nitride.zincblen.gif

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