Question

Derive packing efficiency of

1. ccp and hep structure 

2. Body centered cubic 

3. Simple cubic

Answer 1

1. Packing Eciency in ccp and hep Structures :

In the case of ccp and hep, the edge length

We know that each unit cell in ccp structure has 4 spheres.

Volume of sphere = \(\frac{4\pi r^3}{3}\)

Volume of the cube = a³

^{Packing efficiency = \(\frac{\text(Volume\,occupied\,by\,4\,spheres\,in\,the\,unit\,cell)}{\text(Total\,volume\,of\,the\,unit\,cell)}\times100\)} 

^{= \(\frac{4\times{\frac{4}{3}\pi r^3}}{(2\sqrt2 r)^3}\times100\)%}

^{= 74%}

2. Packing Efficiency of Body Centred Cubic Structures:

In this case radius of a sphere.

We know that bcc has 2 spheres in the unit cell.

∴ Packing efficiency = ^{\(\frac{2\times{\frac{4}{3}\pi r^3}}{[(\frac{4}{\sqrt3} r)]^3}\times100\)%}

^{= 68%}

3. Packing Efficiency is Simple Cubic Lattice : 

In simple cubic lattice edge length ‘a’ and radius of the sphere ‘r’ are related as,

We know that a simple cubic unit cell contains only one sphere.

∴ Packing efficiency =  ^{\(\frac{1\times{\frac{4}{3}\pi r^3}}{(2 r)^3}\times100\)%}

= 52.36%

= 52.4%