## Bravais Crystal Lattice

*by* Ritika

Bravais crystal lattice in crystallography is used to explain the geometrical symmetry of a crystal in details. The other term used to refer to bravais crystal lattice is space lattice.

There are about fourteen distinct bravais crystal lattices. A crystal structure is one of the characteristics of minerals which form the base for bravais crystal lattice in crystallography. Those are the 14 bravais lattices used to describe crystal structures. A bravais lattice is the period arrangement of points that through repeated translation of the lattice vectors will fill space. All crystals could be defined in detail by any of these fourteen bravais crystal lattices. Bravais lattice took its name after Auguste Bravais.

### Lattice Type and Lattice Centering

The kinds of lattice centering used by bravais lattice system are:

- Primitive Centering: In this, lattice points lie on the corner of the cells alone.
- Body Centered: In the body centered lattice centering, there are lattice points lying on the corner of the cells along with one more lattice point placed on the centre of the cell.
- Face Centered: In the face centered style, there are lattice points lying on the corner of the cells along with a lattice point placed at centre of each of the faces of the cell.
- Centered on a Single Face: In this, as the name implies, there is one additional lattice point at the center of one of the cell faces.

The seven crystal systems are combined with the various possible lattice centering defined as above which results in bravais crystal lattice in crystallography. The two dimensional centering is simple since a parallelogram has only one face whereas the three dimensional ones have more options which leads to six different centering arrangements.

### Fourteen Bravais Crystal Lattices

The bravis crystal lattice has fourteen bravais lattices which all occupy the three-dimensional space. These fourteen bravais lattices are defined based on the seven crystal systems. There are 230 space groups with 32 crystallographic point groups. Except quasi crystals, all other crystalline materials lie in one of the fourteen bravais crystal lattices. The fourteen bravais crystal lattices or space lattices are:

- Triclinic Bravais Crystal Lattice
- Simple Monoclinic Bravais Crystal Lattice
- Base Centered Monoclinic Bravais Crystal Lattice
- Simple Orthorhombic Bravais Crystal Lattice
- Base-Centered Orthorhombic Bravais Crystal Lattice
- Body-Centered Orthorhombic Bravais Crystal Lattice
- Face-Centered Orthorhombic Bravais Crystal Lattice
- Hexagonal Bravais Crystal Lattice
- Rhombohedral or Trigonal Bravais Crystal Lattice
- Simple Tetragonal Bravais Crystal Lattice
- Body-Centered Tetragonal Bravais Crystal Lattice
- Simple Cubic or Isometric Bravais Crystal Lattice
- Body-Centered Cubic Bravais Crystal Lattice
- Face-Centered Cubic Bravais Crystal Lattice

Let’s know a little about each of the above fourteen bravais crystal lattices.

#### Triclinic Bravais Crystal Lattice

In the triclinic bravais crystal lattice, vectors of unequal length are used for defining the crystal system. In addition, in this crystal lattice, the three vectors used are not mutually orthogonal.

#### Simple Monoclinic Bravais Crystal Lattice and Base Centered Monoclinic Bravais Crystal Lattice

The monoclinic bravais lattice also is defined by using vectors of unequal length. The resulting structure is a rectangular prism with base having the shape of a parallelogram. In this bravais crystal lattice, two pairs of perpendicular vectors are used with third pair having an angle other than 90 degrees.

#### Simple Orthorhombic Bravais Crystal Lattice , Base-Centered Orthorhombic Bravais Crystal Lattice , Body-Centered Orthorhombic Bravais Crystal Lattice and Face-Centered Orthorhombic Bravais Crystal Lattice

If a cubic lattice is stretched along two lattice vectors it results in a rectangular prism with base having the shape of rectangle and this is termed as orthorhombic lattices. In the orthorhombic lattices all the three bases intersect at 90 degrees with also the three vectors being mutually orthogonal.

#### Hexagonal Bravais Crystal Lattice

In the hexagonal crystal lattice, the symmetry is equal as the right prism has a hexagonal base. An example of this is graphite.

#### Rhombohedral or Trigonal Bravais Crystal Lattice

The rhombohedral bravais crystal lattice is also termed as Trigonal bravais crystal which is well defined by using vectors of equal length. In addition, it is also important to note that all the three vectors used to define the rhombohedral bravais crystal lattice are not mutually orthogonal. The rhombohedral bravais crystal lattice is similar to the cubic system being stretched along diagonally across the body.

#### Simple Tetragonal Bravais Crystal Lattice and Body-Centered Tetragonal Bravais Crystal Lattice

Tetragonal crystal lattice is obtained by using a cubic lattice and stretching the same along a lattice vector. The tetragonal crystal lattice is similar to a rectangular prism having the base as the shape of square.

#### Simple Cubic or Isometric Bravais Crystal Lattice

Each corner of a cube is defined with a lattice point in the case of cubic crystal system. Also, each lattice point shares equal spacing between eight adjacent cubes

#### Body-Centered Cubic Bravais Crystal Lattice

In the case of body-centered cubic crystal system, there are eight corner points defined with a lattice point in each of these right corner points. In addition to this one, more lattice point is used to define the center of the unit cell.

#### Face-Centered Cubic Bravais Crystal Lattice

In the face-centered cubic crystal system, the lattice points are placed on the faces of the cube.

### Bravais lattices in Two Dimensional Space

There are five Bravais lattices in two dimensional spaces. They are:

- Oblique
- Rectangular
- Centered Rectangular
- Hexagonal and
- Square

### Bravais lattices in Four Dimensional Space

There are fifty two bravais lattices in four dimensional space. Of these fifty two bravais lattices, twenty one are primitive and thirty one are centered.

The point group of the bravais lattice is the set of all point operations that leave the lattice invariant. Also, the bravais lattices associated with the same space group are considered the same type, although they are not equivalent. That is, the bravais lattices associated with the same point group as classified as the same crystal system. It is vital to note that a two-dimensional honeycomb do not form a bravais lattice. A bravais lattice is a lattice in which every lattice points have exactly the same environment. That is, the bravais lattice can be spanned by primitive vectors.