Question

Calculate the efficiency of packing in case of metal crystal for face-centred cubic.

Answer 1

A, B, C are the centres of the sphere. As sphere on the face centre is touching the spheres at the corners evidently $AC=4r$ But from right angled triangle ABC

$AC=\sqrt{A{B}^2+B{C}^2}+\sqrt{a^2+a^2}=\sqrt{2}a$

$\therefore \sqrt{2}a=4r$

or $a=\frac{4}{{\sqrt{2}}^r}$

$\therefore$ volume of the unit cell

$=a^3={\left[\frac{4}{\sqrt{2}}r\right]}^3=\frac{32}{\sqrt{2}}{r}^3$

Number of spheres in the unit cell

$=8\times \frac{1}{8}+6\times \frac{1}{2}=4$

volume of 4 spheres $=4\times \frac{4}{3}\pi r^3=\frac{16}{3}\pi r^3$

$\therefore$ fraction occupied, i.e. packing fraction

$=\frac{16\pi r^3/3}{32r^3\sqrt{2}}=\frac{\pi \sqrt{2}}{6}=0.74$

or % occupied $=74\%$