Question

The volume of this $hcp$ unit cell is

• A. $74\%$
• B. $47.6\%$
• C. $32\%$
• D. $26\%$

Answer 1

Answer: (A)

In close packed arrangement, side of the base $=2r$
$\Rightarrow RS=r$
Also MNR is equilateral triangle, $\angle PRS=30^{\circ }$
In triangle $PRS,\cos 30^{\circ }=\frac{RS}{PR}=\frac{\sqrt{3}}{2}$
$\Rightarrow PR=\frac{2}{\sqrt{3}}RS=\frac{2}{\sqrt{3}}r$
In right angle triangle $PQR:PQ=\sqrt{Q{R}^2-P{R}^2}=2\sqrt{\frac{2}{3}}r$
$\Rightarrow$ Height of hexagon $=2PQ=4\sqrt{\frac{2}{3}}r$
$\Rightarrow$ Volume $=$ Area of base $\times$ height $=6\frac{\sqrt{3}}{4}(2r{)}^2\times 4\sqrt{\frac{2}{3}}r$
$=24\sqrt{2}{r}^3$