# Bravais Lattice

Before understanding the Bravais Lattice, you need to have an idea about basic terms and concepts related to the topic. Such as to understand lattices, you must have an idea about crystalline solids. Solids are mainly of two types – amorphous and Crystalline. Amorphous solids do not have a well-defined structure and possess no geometric shapes. While crystalline solids possess geometric shapes and have well-defined structures. In crystalline solids constituent particles are arranged in a specific manner in three dimensional ways. They are made up of three - dimensional repetitive unit cells which have specific patterns.

A lattice is a geometric arrangement of the points in space at which the atoms, molecules, ions or constituent particles of a crystal occur. It describes the arrangement of particles in the crystal. Unit cell of a crystal is defined by lattices. Lattice point is the point or position in the unit cell or on the lattice in a crystal where the probability of finding an atom or ion is the highest.

What is Bravais Lattice?

Bravais Lattice is an infinite array of discrete points in three - dimensional space generated by a set of discrete translation operations. It is named after French physicist Auguste Bravais. He is known for his work in crystallography. He gave the concept of Bravais lattice and formulated Bravais Law.

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 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 combination of 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 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 possess 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 -

For description, we are using a, b and c to denote the dimensions of the unit cells and $\alpha ,\beta ,\lambda$to denote corresponding angles in the unit cells.

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

a = b = c

$\alpha = \beta = \lambda = 9{0^0}$

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

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. For orthorhombic systems -

a $\ne$ b $\ne$ c

$\alpha = \beta = \lambda = 9{0^0}$

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

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. For rhombohedral systems -

a = b = c

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

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

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. For triclinic systems -

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.

14 - Types of Bravais Lattice in Tabular Form

We are giving all 14 Bravais lattices with unit cell structures in tabular form here for comparative study of them and your easy understanding -

 14 -Bravais Lattices Lattice System Primitive (P) Base Centered (C) Body centered (I) Face centered (F) Total Cubic a = b = cα= β=γ=90° - 3 Tetragonal a = b ≠ cα= β=γ=90° - - 2 Orthorhombica ≠ b ≠ cα= β=γ=90° 4 Hexagonal a = b ≠ cα=120°,β=γ=90° - - - 1 Rhombohedrala = b = cα= β=γ≠ 90° - - - 1 Monoclinic - - 2 Triclinic - - - 1 Total 14

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