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Home / Chemistry / Packing Efficiency of a Unit Cell - Learn about Cubic Structures and their Efficiencies

Packing Efficiency of a Unit Cell - Learn about Cubic Structures and their Efficiencies

Last Updated on Mar 12, 2025
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A crystal lattice is composed of numerous unit cells, with each lattice point occupied by a single constituent particle. The unit cell is a three-dimensional structure containing one or more atoms. Regardless of the type of packing, there are always some void spaces in the unit cell. The packing fraction of the unit cell, which is the percentage of spaces filled by the particles, is known as the packing efficiency. In this section, we will explore the concept of packing efficiency and how it is calculated in different types of packing in unit cells.

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Packing efficiency in face centered cubic unit cell

Hexagonal close packing (hcp) and cubic close packing (ccp) have the same packing efficiency. Consider a unit cell of edge length “a”. The length of the face diagonal, b, can be calculated using the Pythagoras theorem.

We can derive the equation for the radius of the sphere, r, and the face diagonal (b) as follows:

 

The volume of the cube and the sphere can be calculated as follows:

Volume of the cube =
Volume of the sphere =
No. of Sphere present in one FCC unit cell = 4
Total volume of the sphere =

 

The packing efficiency is then calculated as:

= 74%

 

Packing efficiency in body centered cubic unit cell

In a body centered cubic unit cell, one atom is located at the center of the body in addition to the corners of the cube. Let us take a unit cell of edge length “a”. The length of the body diagonal, c, can be calculated using the Pythagoras theorem.

The equations for the radius of the sphere, r, length of the body diagonal, c, and edge length of the unit cell, a, can be derived as follows:

 

In a body centered cubic structure, each unit cell has two atoms. The packing efficiency can be calculated as follows:

= 68%

 

Packing Efficiency of Diamond

In the structure of diamond, C atom is present at all corners, all face centres and 50 % tetrahedral voids. The packing efficiency of diamond is 34%, where:

Number of C atoms per unit cell = 8
a = edge length of unit cell

 

Packing efficiency in simple cubic unit cell

In a simple cubic unit cell, atoms are located at the corners of the cube. The radius of the atom can be given as follows:

 

In a simple cubic structure, each unit cell has only one atom. The packing efficiency can be calculated as follows:

= 52.4%

 

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Frequently Asked Questions

The percentage of the total space which is occupied by the particles in a certain packing is known as packing efficiency. The constituent particles i.e. atoms, ions or molecules are closely packed in the crystal lattice. They can do so either by cubic close packing(ccp) or by hexagonal close packing(hcp). In both the cases, a number of free spaces or voids are left i.e, the total space is not occupied.

The Packing efficiency of Hexagonal close packing (hcp) and cubic close packing (ccp) is 74%.

The cubic closed packing is CCP, FCC is cubic structures entered for the face. When we put the atoms in the octahedral void, the packing is of the form of ABCABC, so it is known as CCP, while the unit cell is FCC. These are two different names for the same lattice.

The packing efficiency of body-centred cubic unit cell (BCC) is 68%.

The packing efficiency of simple cubic unit cell (SCC) is 52.4%.
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