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

Using the touching-atom relation for each cell ($a=2r$ SC; $\sqrt3\,a=4r$ bcc; $\sqrt2\,a=4r$ fcc):

• Simple cubic: $\dfrac{\frac43\pi r^3}{(2r)^3}=\dfrac{\pi}{6}=52.4\%$. …

General formula.

1. Simple cubic ($z=1$, atoms touch along the edge $\Rightarrow a = 2r$).

   $\text{PE} = \frac{\frac{4}{3}\pi r^3}{(2r)^3} = \frac{\frac{4}{3}\pi r^3}{8r^3} = \frac{\pi}{6} = 0.524 = 52.4\%$

2. Body-centred cubic ($z=2$, atoms touch along the body diagonal $\Rightarrow \sqrt3\,a = 4r,\ a = \dfrac{4r}{\sqrt3}$). $\text{PE} = \frac{2\cdot\frac{4}{3}\pi r^3}{\left(\frac{4r}{\sqrt3}\right)^3} = \frac{\frac{8}{3}\pi r^3}{\frac{64r^3}{3\sqrt3}} = \frac{8\pi\sqrt3}{64} = \frac{\sqrt3\,\pi}{8} = 0.680 = 68\%$ …

For each cell, write the touching-atom geometry to relate $a$ and $r$ (edge, body-diagonal, face-diagonal), put $z$ atoms …

• Using the wrong touching direction (e.g. body diagonal for fcc instead of face diagonal). …