 # Unit Cell Packing Efficiency

What Is Unit Cell Packing Efficiency?

A Unit Cell may be viewed as a 3-D Structure made up of one or more atoms. Some void space is always present irrespective of the type of packing the cell has. The fraction of total space that is filled with the inherent constituent particles of a particular cell or structure is called the packing fraction. It can be obtained by dividing the total volume occupied by constituent particles by the cell's total volume.

When this is shown as a percentage i.e., out of the total space, the percentage that is held up by constituent particles is called the Packing Efficiency of a Unit Cell.

Packing Efficiency Formula = $\frac{\text{No of Atoms x Volume Occupied by 1 atom}}{\text{Total Volume of Unit Cell}}$ X 100

We can say that Packing Fraction, when multiplied by 100, is seen as a percentage, it becomes the Packing Efficiency of that particular cell.

So, how to find the packing fraction:

Packing Fraction Formula =$\frac{\text{Volume Occupied by all constituent particles}}{\text{Total Volume of Unit Cell}}$

There is always some space inside a cell, and this is known as Void Space. It can be derived as follows:

Void Space Fraction: 1- Packing Fraction

Percentage of Void Space: 100 - Packing Efficiency

Packing Efficiency of a Simple Cubic Crystal Lattice (SCC)

In a simple cubic crystal structure, particles are located only on the corners of the cube. The following relation gives the edge or side length of the cube(a) and radius(r) of constituent particles:

a = 2r

A Simple Cubic Crystal contains only single atom and hence the Volume Occupied by atoms is given as:

The Volume of Cube = a3 i.e. (2r)3 = 8r3

Therefore, Packing Efficiency = $\frac{4\pi r^{3}}{3\times 8r^{3}}$ X 100

It totals to 52.4% while Void Space Percentage is 47.6%

It highlights that a Simple Cubic Crystal Lattice is Loosely Bound.

Packing Efficiency of Body Centred Cubic Lattice

In this kind of structure, the particles are present at the edges, and a single particle is present in the centre as well. We can calculate the packing efficiency as follows:

When Side is ‘a’ and Radius is ‘r’:     r =  $\frac{\sqrt{3}}{4}$ a

It contains two atoms and area occupied by one atom is  $\frac{4\pi r^{3}}{3}$

The volume of the cell in terms of 'r' is:    =  $\frac{64r^{3}}{3\sqrt{3}}$

BCC Packing Efficiency = 68.04%

The packing fraction of BCC is $\frac{68}{100}$.

32% of the total Volume remains void.

Packing Efficiency of Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP)

Both the Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP) structure have the same packing efficiency. The relationship between side represented as 'a' and radius is represented as 'r' is given as:

a = 2$\sqrt{2}$r

These structures are also face-centered cubic lattice and have atoms situated on the eight corners of the cube and the center.

The Volume in terms of 'r' can be, given as:

($2\sqrt{2}r)^{3}$ = 16$\sqrt{2}$$r^{3}$

Packing Efficiency of CCP = $\frac{4\times (\frac{4\pi r^{3}}{3})}{16\sqrt{2}r^{3}}$ X 100 = 74%

Packing fraction of HCP and FCC -  $\frac{74}{100}$

Leaving only 26% Void Space

## Therefore, we can Summarize:

 Type of Structure Number of Atoms ‘a’ and ‘r’ Relationship Packing Efficiency Void Space Scc 1 a = 2r 52.4% 47.6% Bcc 2 r = $\frac{(\sqrt{3})}{4}$a 68.04% 31.96% Hcp and Ccp – Fcc 4 a = $2\sqrt{2}r$ 74% 26%

How To Mathematically Find The Relationship Between 'a' And 'r.'?

In a Simple Cubic Structure:

Since the atoms are only on the corners, radius becomes half the side ,i.e., r = $\frac{a}{2}$

In a Body Centred Cubic Structure:

In this case, since atoms are on the corners and an atom is present in the center, we draw a diagonal, and its length (c) can be calculated using Pythagoras theorem.

We get c = $\sqrt{3}a$

And since radius = 4 X Diagonal (as shown in the figure under Body Centred Cubic Lattice)

r = $4\sqrt{3}a$

In hcp and ccp i.e., Face Centred Cubic Structure

Again, we draw the face diagonal and as shown in the figure,

b = $\sqrt{2}a$

r = $\frac{d\times b}{4}$

It finally gives us: a = 2$\sqrt{2r}$

1. What is the Significance of Packing Efficiency?

Packing Efficiency holds prime importance for three chief reasons:

• It is useful in determining and defining the structure of the solid.

• It affects other attributes such as isotropy, density, and consistency. It tells how well an element is bonded

• It provides an insight into mechanical, chemical, physical etc. and various other properties as well.

2. How are Coordination Numbers Related to Packing Efficiency?

In simple terms, Coordination number refers to the number of atoms a unit cell is touching, and we are aware that the unit cell is the smallest representation of an entire crystal. Another important point is that the stability of a solid is generally accepted when it has a high packing efficiency and a high coordination number. Here are the coordination numbers of different cubic structures:

• BCC – 8

• HCP – 12

• CCP – 12

• Simple Lattice - 6

3. What Does Increase or Decrease in Packing Efficiency Denote?

Packing efficiency helps in understanding how low the void space is and how closely the atoms are bonded. It essentially means that the larger the packing Efficiency, the more stable and more closely bonded a solid is meant to be. It also portrays how stable the solid is if it is closely bound. Low packing efficiency signifies large void space, and hence the outcome is a loosely bonded substance. Therefore, as per the packing efficiencies of different cubic structure, we can establish the following hierarchy according to their packing efficiencies:

Fcc (74%) > Bcc (68.04%) > Scc (52.4%)