Packing Efficiency: Structure, Types & Diagram

Packing Efficiency is the percentage of total space in a unit cell that is filled by the constituent particles, such as atoms, ions, or molecules, packed within the lattice. It is the total amount of space occupied by these particles in three-dimensional space. Simply, it can be understood as the specified percentage of the total volume of a solid which is occupied by spherical atoms. The formula for packing efficiency is, 

Packing Efficiency = \(\frac{Total\ Volume\ of\ Sphere}{Total\ Volume\ of\ Cube} \times 100\)

Packing Efficiency can be evaluated through three different structures of geometry which are:

• Cubic Close Packing (CCP) and Hexagonal Close Packing (HCP).
• Body-Centred Cubic Structures (BCC)
• Simple Lattice Structures of Cubic

Key Terms: Unit Cell, Atoms, Ions, Cubic Structures, Lattice, Packing Efficiency, Lattice, Body-Centered Cubic Structure, Simple Lattice

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Factors that determine Packing Efficiency

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The factors that determine the packing efficiency of a unit cell are:

1. The number of atoms in a lattice structure
2. The volume of a unit cell
3. The volume of atoms

The video below explains this:

Packing efficiency Detailed Video Explanation:

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Structures of Packing Efficiency

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The packing efficiency of different lattices is as given below.

Packing Efficiency of HCP & CCP Structures

Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP) are equally efficient. They have the same packing efficiency.

• In Hexagonal Close Packing (HCP), the alternating layers cover each other’s gap.
• The spheres in one layer line up with the gap of the previous layer.
• In Cubic Close Packing (CCP), the layers are exactly placed above each other in symmetry.
• The layers when placed form a cube. 

The packing efficiency in a Cubic Close Packing (CCP) structure can be demonstrated as follows –  

In the above diagram, let ‘A’ be the edge length of the unit cell and AC, which is also equal to b, be the face diagonal. 

While looking at the face ABCD of the cube, we can see a triangle is formed. Let r be the radius of each sphere. We proceed by correlating the radius and the edge of the cube.

In triangle ABC,

AC² = BC² + AB²

Since AC= b and BC = AB= a, 

We get

b² = a² + a² = 2a²

b = √2 a .....(i)

As the radius of each sphere is r, we can rewrite the equation as

b = 4r …..(ii)

We can write from (i) and (ii)

a = 2√2 r or 4r = √2a

Since, the volume of one sphere =  4/3 πr³ 

And we know that there are four spheres in a Cubic Close Packing (CCP) structure. 

The total volume of four spheres is therefore equivalent to 4 × 4/3 πr³ 

Total Volume of a cube is (edge length)³ i.e, ( a³) or in terms of r it is (2√2 r)³

Packing Efficiency = \(\frac{Total\ Volume\ of\ Sphere}{Total\ Volume\ of\ Cube} \times 100\)

= \(\frac{4 \times (4/3) \pi r^3}{(2 \sqrt{2} r)^3}\)

= 74%

Body-Centred Cubic Structures

Three atoms are arranged diagonally in Body- Centred Cubic Structures.

According to the Pythagoras theorem, in triangle EFD,

b² = a² + a²

b² = 2a²

b = √2 a

According to the Pythagoras theorem, now in triangle AFD, 

c² = a² + b²

= a² + 2a² = 3a²

c = √3 a

As the radius of each sphere is r, we can rewrite the equation as

c = 4r

√3 a = 4r

r = √3/4 a

In the Body- Centred Cubic (BCC) Structures, there are two atoms, so the volume of constituent spheres will be: 2 × (4/3) π r³

Packing Efficiency = \(\frac{Total\ Volume\ of\ Sphere}{Total\ Volume\ of\ Cube} \times 100\)

= \(\frac{2 \times (4/3) \pi r^3}{(4/ \sqrt{3} r)^3}\)x 100

= 68%

Metals like chromium and iron fall under Body-Centred Cubic Structures.

Simple Lattice Structures

The atoms are located only on the corners of the cube in a simple cubic lattice. Along the edge, the particles touch each other. 

Simple Lattice Structures

In the above figure, we can see that only at the edges particles are in touch. Let the radius of each atom be ‘r’ and the edge length of the cube be ‘a’ we can co-relate them as

2a = r

Since only 1 atom contains a simple cubic unit, the volume of the unit cell with occupied one atom will be

4/3 π r³

The volume of the unit cell is

( a³) or ( 2a)³ that is, 8a³

Packing Efficiency = \(\frac{Total\ Volume\ of\ Sphere}{Total\ Volume\ of\ Cube} \times 100\)

= \(\frac{ (4/3) \pi r^3}{(8 r)^3}\)x 100

= 52.4%

For example, metals like lithium and calcium.

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Importance of Packing Efficiency

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Packing Efficiency is important as:

• Packing Efficiency represents the solid structure of the object.
• It displays different properties of solids like consistency, density, and isotropy.
• With the help of Packing Efficiency, different attributes of solid structures can be derived.

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Types of Cubic Structures

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The packing efficiency of different solid structures are as tabulated below.

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Things to Remember

• Packing Efficiency is defined as the percentage of total space in a unit cell that is filled by the constituent particles within the lattice.
• Packing Efficiency can be assessed in three structures – Cubic Close Packing and Hexagonal Close Packing, Body-Centred Cubic Structures, and Simple Lattice Structures Cubic.
• For a Simple Cubic cell, the number of atoms in the unit cell is 1 and the coordination number is 6.
• For Body-centred cubic cell, the number of atoms in the unit cell is 2 and the coordination number is 8.
• For Face centered cubic cell, the number of atoms in the unit cell is 4 and the coordination number is 12.