Question

What is the packing efficiency for end centred unit cell?

Answer 1

In end centered unit cell the atoms are present on the corners and on the centers of any two opposite faces. So, effective number of atoms in unit cell $=\frac{1}{8}\times 8+\frac{1}{2}\times 2=1+1=2$

Now, at one of the faces toms along face diagonals are touching each other. If $a$ is the edge length of cube and $r$ is the radius of the atom.

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

So, volume occupied $=2\times \frac{4}{3}\times \pi \times r^3$

Total volume $=a^3={\left(\frac{4r}{\sqrt{2}}\right)}^3=\frac{64}{2\sqrt{2}}{r}^3=\frac{32}{\sqrt{2}}{r}^3$

So, $packingefficiency=\frac{volumeoccupied}{totalvolume}\times 100$%

$=\frac{\frac{4}{3}\times \pi \times r^3\times 2}{\frac{3}{2}\sqrt{2}\times r^3}\times 100$%

$=37.02$%