Question

Calculate packing efficiency in face-centred cubic lattice.

Answer 1

Step 1 : 

Radius of sphere : In the unit cell of face-centred cubic lattice, there 8 atoms at 8 corners and 6 atoms at 6 face centres. 

Consider the face ABCD.

The atoms are in contact along the face diagonal BD. 

Let a be the edge length and r, the radius of an atom. 

Consider a triangle BCD. 

BD² = BC² + CD²

= a² + a² = 2a

∴ BD = √2a

From figure, 

BD = 4r

Step 2 : 

Volume of sphere :

Step 3 : 

Total volume of particles : The unit cell of fee crystal lattice contains 4 particles

.∴ Volume occupied by 4 particles = 4 x \(\frac{\pi a^3}{12\sqrt2}\)

= \(\frac{\pi a^3}{3\sqrt2}\)

Step 4 : 

Packing efficiency :

∴ Packing efficiency = 74% 

∴ Percentage of void space = 100 – 74 

= 26%

Edge length and particle parameters in cubic system :

Coordination number and packing efficiency in systems :

Lattice	Coordination number of atoms	Packing efficiency
1. scc	6 : four in the same layer, one directly above and one directly below	52.4%
2. bcc	8 : four in the layer below and four in the layer above	68%
3. fcc/ccp/hcp	12 : six in its own layer, three above and three below	74%