Question

In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell arc regular hexagons and three atoms are sandwiched between them. A space-filling model of this structure, called hexagonal close packed (hcp), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of these three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be 'r'.

The volume of this hcp unit cell is:

• A. $24\sqrt{2}{r}^3$
• B. $16\sqrt{2}{r}^3$
• C. $12\sqrt{2}{r}^3$
• D. $\frac{64}{3\sqrt{3}}{r}^3$

Answer 1

The correct option is A $24\sqrt{2}{r}^3$

Effective number of unit cell (Z)$=[1\times \frac{1}{2}+6\times \frac{1}{6}]\times 2+3\times 1=6$
$\Rightarrow Z=6,a=2r$
$\text{Volume of hexagonal unit cell}=6\times \frac{\sqrt{3}}{4}{a}^2\times h$
$\text{Since,}\frac{\frac{h}{2}}{a}=\frac{\sqrt{2}}{\sqrt{3}}$
$\therefore h=4r\frac{\sqrt{2}}{\sqrt{3}}$
$\Rightarrow \text{Volume of hexagonal unit cell}=6\times \frac{\sqrt{3}}{4}(2r{)}^2\times 4r\frac{\sqrt{2}}{\sqrt{3}}$
$\Rightarrow \text{Volume of hcp unit cell}=24\sqrt{2}{r}^3$