What are the packing fraction $(P . F)$ of Diamond cubic and Hexagonal close packing

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0.34,0.74

0.52,0.34

0.69,0.52

0.74,0.34

answer is D.

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Detailed Solution

Diamond cubic $(D C)$

Radius of atom, $r = \frac{\sqrt{3} . a}{8}; Z_{e f f} = 8$

$P F = \frac{8 \times \frac{4}{3} π r^{3}}{a^{3}} = \frac{8 \times \frac{4}{3} π {(\frac{\sqrt{3} . a}{8})}^{3}}{a^{3}} = \frac{\sqrt{3} π}{16} = 0.34$

$h c p; r a d i u s o f a t o m r = \frac{a}{2}; z_{e f f} = 6$

$\begin{array}{l} h e i g h t = 4 r \times \sqrt{\frac{2}{3}}; B a s e a r e a = 6 \sqrt{3} . r^{2} \\ v o l u m e o f u n i t c e l l = 24 \sqrt{2} r^{3} \end{array}$

$\begin{array}{l} P . F = \frac{6 \times \frac{4}{3} π r^{3}}{a^{3}} = \frac{6 \times \frac{4}{3} π r^{3}}{24 \sqrt{2} r^{3}} \\ = \frac{π}{3 \sqrt{2}} = 0.74 \end{array}$