## The cubic unit cell of a crystal lattice

The idea about cubic lattice developed from the internal regularity of the structure of the **cubic crystal**. Thus a highly ordered three-dimensional structure formed by atom, ions or molecules is called a cubic **crystal lattice**.

Consider a cube in which a lattice point arranged in a regular and repeated manner in the center of each face as well as at the corner. This unit cell can be represented to give a space lattice.

The space lattice divided into a large number of small symmetrical units when similar points connected by sets of parallel lines along the three coordinate axes. These basic units of the space lattice are known as unit cells.

### Bravais lattice definition

Bravis showed that there are 14 different unit cells to account for the lattice points at the corner as well as these at the centers and on the some of the faces.

These 14 unit cells originated from seven crystal systems are known as the Bravais lattice. Among these 14 crystal systems cubic system has three Bravais lattices

- Primitive or simple (P)
- Body-centered (I)
- Face centered (F)

#### Number of atoms in the simple cubic unit cell

In simple or primitive cubic lattice where atoms are present at the corners only. Thus each atom equally shared by eight atoms because each cube contains eight corners.

Hence the contribution of each atom to the unit cell = 1/8. So the number of atom per unit cell = 8 × (1/8) = 1 only.

#### Number of atoms in the body-centered cubic unit cell

In body-centered cubic lattice where the atoms are present at the body and the corner of the cube.

But the body-centered atoms belong exclusively to the unit cell. Thus the total number of atoms per unit cell = 1 (for the body) + 1 (for corner) = 2.

#### Number of atoms in a face-centered cubic unit cell

In a face-centered cubic lattice where the atoms are present at the six faces and the eight corners of the cube.

But face atoms equally shared by two unit cells. Thus the total number of atoms per unit cell = 3 (for face) + 1 (for corner) = 4.

### Law of rational indices and miller indices

In order to describe the structure of a crystalline solid, we need to discuss the orientation of the plane passing through the lattice point of the crystal. But the orientation of the lattice plane describes by the intercepts of the plane on the three basis vector of the lattices.

Thus the intercepts of any plane along the three crystallographic axes either equal to the ratio of the unit cell or some integral multiple of them. This is known as the law of rotational indices.

If a plane cut two three dimensional axes by a and b and third axis in infinity than the Weiss indices = 1, 1, ∞.

But in the miller system of the indexing plane, we avoided ∞ because it is rather inconvenient. Thus in this notation, reciprocal of Weiss indices taken and then reciprocal are converted into a small set of integers.

Planes | Weiss indices | Miller indices |

Set A | 1,1,∞ | 110 |

Set B | 1,2,∞ | 1,½,0 or 210 |

Set C | 1,∞,∞ | 100 |

### Lattice Planes in the cubic crystal system

A cubic system is the simplest type of crystal system. Thus the intercept on the three-axis is equal and all the angles equal to 90^{0}.

In the simple cubic lattice, the points are at the corner of the cube. Thus the planes that can pass through the lattice point have miller indices (100), (110), (111).

- The (100) planes cut only on the x-axis but are parallel to y and z-axis.
- (110) planes cuts x and y-axis but parallel to the z-axis.
- (111) planes cuts x, y, z-axis.

The distance between the two adjacent planes of a cubic crystal determined with the help of Bragg’s X-ray deflection measurement.