Question

Calculate the efficiency of packing in case of a metal crystal for :
(i) Simple cubic
(ii) body centred cubic
(iii) Face centred cubic

Answer 1

(i) Simple cubic
Radius of sphere = r
Edge length = a
As simple cubic structure is formed by one sphere hence,
the volume of one sphere = \(\frac { 4 }{ 3}\)πr³
As two spheres are touching each other hence,
a = 2 r

No. of spheres per unit cell = \(\frac { 1 }{ 8}\) = 1

Volume of cube = (a)³= (2r)³ = 8r³

= 52.4%

(ii) Packing efficiency in body centred cubic structure (bcc) .
Radius of sphere = r.
Edge length = a
As bcc structure is formed by two spheres.

It we see the arrangement of spheres in AD then

Put the value of AD in equation (i)
4r = a√3
a = \(\frac { 4r }{ \sqrt { 3 } } \)

(iii) Packing efficiency in hcp or ccp or foc structure
Radius of sphere = r.
Edge length = a
Volume of one cube = \(\frac { 4 }{ 3}\)πr³
Since fcc is formed by 4 sphere, so the volume of four spheres = 4 x 4 \(\frac { 4 }{ 3}\)πr³
= \(\frac { 16 }{ 3}\)πr³

If we consider AC then.

AC = 4r
Put the value of AC in eq (i).