Unit Cell Packing Efficiency in Crystal Structures

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)

(Image will be Uploaded Soon)

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

(Image will be Uploaded Soon)

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)

(Image will be Uploaded Soon)

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:

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\[\sqrt{2r}\] 

Unit Cell Packing Efficiency

Unit cell can be defined as a three-dimensional structure that is made of one or more than one atom. Even when there is packing in the cell, a certain void is present in it. The space is filled by other constituents or particles. The fraction of total space that is filled with the particular cell or structure is called the packing fraction. This can be obtained by dividing the volume of the constituent particles filled in the space by the total volume of the cell. When it is represented as a percentage then the percentage of the space applied by the constituent particles out of the total space present in the structure is called the packing efficiency of the unit cell.

A lattice is largely made of a number of unit cells in which the lattice point is filled or occupied by a constituent particle. This unit cell of the lattice is a three-dimensional structure that has one or more atoms and also void spaces irrespective of the packing present. The cubic closed packed or ccp and the hexagonal closed packed or hcp are two efficient lattices when we consider packing. The packing efficiency of both of these is 74% which means 74% of the space is filled. For a simple cubic lattice, the packing efficiency is 52.4% and the packing efficiency is 68% for a body-centered cubic lattice or bcc.