Unit Cell

What is a Unit Cell?

The smallest repeating unit of the crystal lattice is the unit cell, the building block of a crystal.

The unit cells which are all identical are defined in such a way that they fill space without overlapping. The 3D arrangement of atoms, molecules or ions inside a crystal is called a crystal lattice. It is made up of a large number of unit cells. One of the three constituent particles takes up every lattice point.

A unit cell can either be primitive cubic, body-centered cubic (BCC) or face-centered cubic (FCC). In this section, we will discuss the three types of the unit cell in detail.

Types of Unit Cell

A large number of unit cells together make a crystal lattice. Constituent particles like atoms, molecules are also present. Each lattice point is occupied by one such particle.

  1. Primitive Cubic Unit Cell
  2. Body-centered Cubic Unit Cell
  3. Face centered cubic unit cell
Types of Unit Cell

Types of Unit Cell

1. Primitive Cubic Unit Cell

In the primitive cubic unit cell, the atoms are present only at the corners. Every atom at the corner is shared among 8 adjacent unit cells. There are 4 unit cells in the same layer and 4 in the upper (or lower) layer. Therefore, a particular unit cell has the only 1/8th of an atom. Each small sphere in the following figure represents the center of a particle that occupies that particular position and not its size. This structure is known as an open structure.

  1. The atoms in the primitive cubic unit cell are present only at the corners
  2. Every atom at the corner is shared among eight adjacent unit cells
  3. Four unit cells are present in the same layer
  4. Four unit cell in the upper/lower layer
  5. Therefore a particular unit cell has the only 18th of an atom
  6. Each small sphere in the following figure represents the center of a particle which occupies that particular position and not its size

Below is an open structure

simple cubic unit cell

simple cubic unit cell

In each cubic unit cell, there are 8 atoms at the corners. Therefore, the total number of atoms in one unit cell is

8 × 1/8 = 1 atom.

2. Body-centered Cubic Unit Cell (BCC)

A BCC unit cell has atoms at each corner of the cube and an atom at the center of the structure. The diagram shown below is an open structure. According to this structure, the atom at the body center wholly belongs to the unit cell in which it is present.

  1. In BCC unit cell every corner has atoms.
  2. There is one atom present at the center of the structure
  3. Below diagram is an open structure
  4. According to this structure atom at the body centers wholly belongs to the unit cell in which it is present.

 

Body centered Cubic Unit Cell

Body centered Cubic (BCC) Unit Cell

Number of Atoms in BCC Cell:

Thus, in a BCC cell, we have:

  • 8 corners × 1/8 per corner atom = 8 × 1/8 = 1 atom
  • 1 body center atom = 1 × 1 = 1 atom

Therefore, the total number of atoms present per unit cell = 2 atoms.

3. Face-centered Cubic Unit Cell (FCC)

An FCC unit cell contains atoms at all the corners of the crystal lattice and at the center of all the faces of the cube. The atom present at the face-center is shared between 2 adjacent unit cells and only 1/2 of each atom belongs to an individual cell.

  1. In FCC unit cell atoms are present in all the corners of the crystal lattice
  2. Also, there is an atom present at the center of every face of the cube
  3. This face-center atom is shared between two adjacent unit cells
  4. Only 12 of each atom belongs to a unit cell

The diagram shown below is an open structure.

face centered cubic unit cell

face centered cubic unit cell

Number of Atoms in BCC Cell

a) 8 corners × 18 per corner atom = 8 × 18 = 1 atom
b) 6 face-centered atoms × 12 atom per unit cell = 3 atoms

Hence the total number of atoms in a unit cell = 4 atoms

Thus, in a face-centered cubic unit cell, we have:

  • 8 corners × 1/8 per corner atom = 8 × 1/8 = 1 atom
  • 6 face-centered atoms × 1/2 atom per unit cell = 3 atoms

Therefore, the total number of atoms in a unit cell = 4 atoms.

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