Calculate the efficiency of packing in case of a metal crystal for

(i) Simple cubic

(ii) Body - centred cubic

(iii) Face-centred cubic (with the assumptions that atoms are touching each other).

(i) In simple cubic, Packing efficiency

= $\frac{Volumeofoneatom\times 100}{Volumeofunitcell}$

= $\frac{\frac{4}{3}\pi r^3}{8r^3}\times 100=\frac{\pi }{6}\times 100\%=52.4\%$

where, r = radius of one sphere or atom.

(ii) In body centred cubic, Packing efficiency

= $\frac{Volumeoccupiedby2atomsintheunitcell\times 100}{Volumeoftheunitcell}$

= $\frac{2\times \left(\frac{4}{3}\right)\pi r^3\times 100}{\left[\right(\frac{4}{\sqrt{3}})r{]}^3}$

= $\frac{\left(\frac{8}{3}\right)\pi r^3\times 100}{\frac{64}{(3\times \sqrt{3})r^3}}=68\%$

(iii) In face centred cubic, packing efficiency

= $\frac{Volumeoccupiedby4atomsintheunitcell\times 100}{Volumeoftheunitcell}$

= $\frac{4\times \left(\frac{4}{3}\right)\pi r^3\times 100}{(2\sqrt{2}r{)}^3}=\frac{\left(\frac{16}{3}\right)\pi r^3\times 100}{16\sqrt{2}{r}^3}=74\%$