# Lattice Type

# Bravais Lattices

In order to study the properties of any crystal, it is necessary to have some knowledge of crystallography. The crystal structure can be regarded as the lattice structure plus a basis. At present, millions of crystals are known, and each crystal has a different natural. Fortunately, there are only 14 Bravais lattices in three-dimensional Euclidean space. The following table is a classification of the 14 Bravais lattices.

In the table above, each graph is a unit cell, the simplest repeating unit in a crystal. According to the periodicity, one may use the primitive cell to replace the unit cell in the calculation. The primitive cell is a minimum volume cell that can be spanned by the primitive lattice vectors **a1**, **a2**, **a3**. The constants * n1*,

*, and*

**n2***denote the numbers of grid points in*

**n3***,*

**x***, and*

**y***directions, respectively.*

**z**## Cubic System

A unit cell of the cubic system is cubic, it is a common lattice structure of minerals. The cubic system can be subdivided into three categories : simple(primitive) cubic, body-centered cubic, and face-centered cubic. The unit cell of lattice in cubic system is a cubic, and the lattice constants satisfy * a = b = c* and α

*β*

**=***γ*

**=***π*

**=***.*

**/2**### (1) Simple Cubic Lattice

For the simple cubic lattice, the unit cell is a primitive unit cell, and we do not have any revision on it. The lattice vectors are of the form

### (2) Face-Centered Cubic Lattice

The standard primitive lattice vectors of FCC are of the form

### (3) Body-Centered Cubic Lattice

The standard primitive lattice vectors of BCC are of the form

## Hexagonal system

The unit cell of lattice in hexagonal system is a right rhombic prism with a parallelogram in the bottom, the rhombic with length * a = b* and included angle of γ

*π*

**=2***, the height of the prism is*

**/3****c**. The unit cell is a primitive unit cell and the standard primitive lattice vectors are of the form

## Rhombohedral system

The Rhombohedral lattice is usually de ned as the rhombohedrally hexagonal cell. The unit cell is consist of the unit cell of hexagonal lattice and two lattice points which located on the longest body diagonal. One may also consider the primitive rhombohedral lattice with lattice constants * a = b = c* and α

*β*

**=***γ*

**=***≠ π*

*. The standard primitive lattice vectors are of the form*

**/2**## Tetragonal system

The unit cell of lattice in tetragonal system is a rectangular prism with a square base, and the lattice constants satisfy and * a=b*≠

*and α*

**c***β*

**=***γ*

**=***π*

**=***.*

**/2**### (1) Primitive Tetragonal

The unit cell of the primitive tetragonal lattice is also a primitive unit cell, and we do not have any revision on it. The lattice vectors are of the form

### (2) Body-Centered Tetragonal

The standard primitive lattice vectors of body-centered tetragonal lattice are of the form

## Orthorhombic system

The unit cell of lattice in orthorhombic system is a rectangular prism with a rectangle base, and the lattice constants satisfy* a*≠

*≠*

**b***and α*

**c***β*

**=***γ*

**=***π*

**=***. W.L.O.G., we may assume that*

**/2***.*

**a > b > c**### (1) Primitive Orthorhombic

It is a primitive unit cell in this case, and we also do not have any revision on it. The lattice vectors are of the form

### (2) Base-Centered Orthorhombic

There are two kinds of base-centered orthorhombic lattice: A-base-centered and C-base-centered, in 3D Euclidean space.

- A-base-centered orthorhombic lattice: The standard primitive lattice vectors of A-bace-centered orthorhombic lattice are of the form

- C-base-centered orthorhombic lattice: The standard primitive lattice vectors of C-bace-centered orthorhombic lattice are of the form

### (3) Face-Centered Orthorhombic

The standard primitive lattice vectors of face-centered orthorhombic lattice are of the form

### (4) Body-Centered Orthorhombic

The standard primitive lattice vectors of body-centered orthorhombic lattice are of the form

## Monoclinic system

The unit cell of lattice in monoclinic system is a prism with a parallelogram in the bottom, and the lattice constants satisfy * a*≠

*, α*

**b***β*

**=***π*

**=***, and γ≠ π*

**/2***. W.L.O.G., we may assume that*

**/2***.*

**a > b**### (1) Primitive Monoclinic

The standard primitive lattice vectors of primitive monoclinic lattice are of the form

### (2) A-Base-Centered Monoclinic

There is only one type of base-centered lattice in monoclinic system: A-base-centered. The standard primitive lattice vectors of A-base-centered monoclinic lattice are of the form

## Triclinic system

The unit cell of lattice in monoclinic system is a parallelepiped with different length, and the lattice constants satisfy* a*≠

*≠*

**b***. In addition, any two of α , β , and γ will not be π*

**c***simultaneously. W.L.O.G., we assume that*

**/2***and*

**a > b > c***γ)*

**b*cos(***α*

**> c[cos(***β*

**)-cos(***γ*

**)cos(***γ*

**)]/sin(***. The standard primitive lattice vectors of triclinic lattice are of the form*

**)**