Packing Fraction

When the atoms are joined together to form a crystal lattice, then they do not perfectly fit into each other. Definitely, some gaps are created between the two atoms. These gaps are known as voids.

There are different ways of packing. We calculate the packing fraction and packing efficiency in order to find out which packing is better. This article focuses on the various packing fractions in detail.

What is a Crystal Lattice?

The concept of the crystal lattice is studied under physical chemistry. The crystal lattice is the three-dimensional arrangement of the atoms or molecules in a regular pattern. The geometric shape of the crystal is highly symmetrical.

For example, the crystals of sodium chloride are arranged in a crystal lattice.

What are Unit Cells?

The repeating unit of a crystal lattice is known as the unit cells. Like a brick is used in making a house. Similarly, a unit cell is used in the formation of a crystal system. The unit cells are placed at the lattice points in the crystal and result in the formation of the crystal structure.

What are the Different Types of Unit Cells?

The unit cells are classified into two broad categories. They are described in detail below:

1. Primitive Unit Cell: In a primitive unit cell, the atoms are arranged only at the corners of the crystal lattice. An example of a primitive unit cell is a simple cubic unit cell.
2. Non-primitive Unit Cell: In a non-primitive unit cell, the atoms are arranged at the corners as well as the other lattice points in a crystal lattice. Examples of non-primitive unit cells are body-centred cubic unit cells (BCC) and face-centred cubic unit cells (FCC).

Simple Cubic Unit Cell

In a Simple Cubic Unit Cell, the atoms are present only at the corners of the crystal lattice. Four unit cells are present in each layer. Eight atoms are present corners, and each atom contributes 1/8 to the crystal lattice.

Since each atom at the corners contributes 1/8 to the crystal lattice, the number of atoms per unit cell is given by 1/8 * 8 = 1.

Body-Centred Cubic Unit Cell

In a Body-Centred Cubic Unit Cell, the atoms are present at the corners as well the body centre of the crystal lattice. Four unit cells are present in each layer. Eight atoms are present at the corners, and one atom is present at the body centre of the crystal lattice. Each atom at the corners contributes 1/8 to the crystal lattice, and the atom at the body centre contributes 1 to the crystal lattice.

Therefore, the number of atoms per unit cells is given by (1/8 * 8) + 1 = 2.

Face-Centred Cubic Unit Cell

In a Face-Centred Cubic Unit Cell, the atoms are present at all the corners as well as the centre of each face of the crystal lattice. Four unit cells are present in each layer. Eight atoms are present at the corners, and 6 atoms are present at the face centres of the crystal lattice. Each atom at the corners contributes 1/8 to the crystal lattice, and the atom at the face centre contributes 1/2 to the crystal lattice.

Therefore, the number of atoms per unit cells is given by

(1/8 * 8) + (1/2 * 6) = 4.

What Do You Mean by Packing Fractions in a Crystal Lattice?

Packing Fraction is the ratio of the volume occupied by the atoms of the crystal to the total volume of the cubic lattice. It tells how much space is occupied by the atoms and how much space is filled with voids. Lower the voids, higher the packing fraction.

Packing Fraction of a Body-Centred Cubic Unit Cell

In a BCC crystal, the relation between the atomic radius and the edge length of the cubic lattice is 4r = (3)1/2a.

The number of atoms per unit cell is 2.

Packing Fraction = [2*(4/3)*3.14*r³] / (a)³

= 68% (approximately).

Packing Fraction of a Face-Centred Cubic Unit Cell

In an FCC crystal, the relation between the atomic radius and the edge length of the cubic lattice is 4r = (2)1/2a.

The number of atoms per unit cell is 4.

Packing Fraction = [4 * (4/3)*3.14*r³] / (a)³

= 74% (approximately)

Conclusion

This article focuses on the crystal structure and its efficiency and packing fraction. It starts from the basics of the crystal structure, diving deep into topics such as different forms of unit cells.

By going through this article, one may be able to understand properly why packing fraction is necessary. We hope that this article proved to be useful.