Bravais Lattice in Crystallography and Solid State Chemistry

The concept of lattice comes along with the concept of crystals. Crystalline solids have definite patterns which arise due to the definite patterns in which the different atoms of the crystals are placed. The definite geometric shapes of crystals are possible due to the formation of a lattice with a series of atoms arranged in that specific pattern to give a well-designed three-dimensional structure. The repetitive pattern of the lattice units forms the actual crystal. The atoms can also be substituted with ions or molecules. Lattice points are the points of finding the constituent atoms of the crystal.

Now when we can understand what is a lattice in a crystal, we can also understand what is braves lattice. Bravais lattice actually denotes all the 14 types of three-dimensional patterns in which the atoms can arrange themselves to form a crystal named after the great physicist Auguste Bravais of France. His work including Bravais laws is an important breakthrough in the field of crystallography. 

Bravais lattices are possible both in two-dimensional and three-dimensional spaces where the lattices are filled without any gaps.

In three-dimensional space, 14 Bravais lattices are there into which constituent particles of the crystal can be arranged. These 14 Bravais lattices are obtained by combining lattice systems with centering types. 

A Lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. 14 Bravais lattices can be divided into 7 lattice systems -

• Cubic 

• Tetragonal 

• Orthorhombic 

• Hexagonal 

• Rhombohedral 

• Monoclinic 

• Triclinic 

Centering types identify the locations of the lattice points in the unit cell. 

Primitive Unit Cell (P) - In this lattice points are found on the cell corners only. It is also sometimes called a simple unit cell. In these constituent particles are found at the corners of the lattice in the unit cell, no particles are located at any other position in the cell. Thus, a primitive cell has only one lattice point. 

Non - Primitive Unit Cells - In these unit cells particles are found in other positions of the lattices as well with corners. These can be divided into the following types -

• Body-Centered (I) - In this lattice points are found on the cell corners with one additional lattice point at the center of the cell. Thus, it has particles at the corners and center of the body or cell. 

• Face Centered (F) - In this lattice points are found on the cell corners with one additional lattice point at the center of each face of the cell. Thus, it has particles at the corners and center of each face.

• Base Centered (C) - In this lattice points are found on the cell corners with one additional lattice point at the center of each face of one pair of parallel faces of the cell. It is also called end-centered. Thus, it has particles at the corners and one particle at the center of each opposite face. 

Not all combinations of lattice systems and centering types give rise to new possible lattices. After combining them, several lattices we get are equivalent to each other. 

14 - Types of Bravais Lattice

All 14 Bravais Lattices show few similar characteristics which are listed below-

• Each lattice point represents one particle of the crystal.

• This constituent particle of the crystal can be an atom, ion, or molecule.

• Lattice points of the crystal are joined by straight lines. 

• By joining the lattice point of the crystal, we get the geometrical shape of the crystal.

• Each one of the 14 Bravais lattices possesses unique geometry. Equivalent lattices have been already excluded which we got after combining lattice systems and centering types. 

List of 14 - Types of Bravais Lattices -

• Cubic - Cubic system shows three types of Bravais lattices - Primitive, base centered and face centered. 

a = b = c   

\[\alpha  = \beta  = \lambda  = 9{0^0}\]

• Tetragonal - Tetragonal system shows two types of Bravais lattices - Primitive, body centered. 

a = b \[ \ne \]c

\[\alpha  = \beta  = \lambda  = 9{0^0}\]

• Orthorhombic - Orthorhombic system shows four types of Bravais lattices - Primitive, body centered, base centered and face centered.

a \[ \ne \] b \[ \ne \] c

\[\alpha  = \beta  = \lambda  = 9{0^0}\]

• Hexagonal - Hexagonal system shows one type of Bravais lattice which is Primitive. 

a = b \[ \ne \]c

\[\alpha  = 12{0^o}      \beta  = \lambda  = 9{0^o}\]

• Rhombohedral - Rhombohedral system shows one type of Bravais lattice which is Primitive. 

a = b = c

\[\alpha  = \beta  = \lambda  \ne 9{0^o}\]

• Monoclinic - Monoclinic system shows two types of Bravais lattices - Primitive, base centered. 

a = b \[ \ne \]c

\[\alpha  \ne 9{0^o}\beta  = \lambda  = 9{0^o}\]

• Triclinic - Triclinic system shows one type of Bravais lattice which is Primitive.

a \[ \ne \]b \[ \ne \]c

\[\alpha  \ne \beta  \ne \lambda  \ne {90^o}\]

Thus, from the cubic system - two, from tetragonal - two, from orthorhombic - four, from hexagonal - one, from rhombohedral - one, from monoclinic two and from triclinic one Bravais lattices are found. If you add all these Bravais lattices, you get a total 14 Bravais lattices.