Question

Packing fraction of an identical solid sphere is $74\%$ in:

• A. simple cubic structure
• B. fcc structure
• C. hcp structure
• D. bcc structure

Answer 1

Answer: (B), (C)

hcp structure
fcc structure

$4R=\sqrt{2}a\Rightarrow R=\frac{\sqrt{2}}{4}a$

Packing factor=$\frac{V_s}{V_e}=\frac{4\times \frac{4\pi }{3}{R}^3}{a^3}=\frac{4\times \frac{4\pi }{3}\times (\frac{\sqrt{2}}{4}a{)}^3}{a^3}$

$=\frac{\sqrt{2}\pi }{6}=74.05\%$

$e=\sqrt{\frac{8}{3}}a,R=\frac{a}{2}$

Packing factor $=\frac{V_s}{V_3}=\frac{6\times \frac{4\pi }{3}{R}^3}{6\times \frac{1}{2}\times \frac{\sqrt{3}}{2}{a}^2\times c}=\frac{6\times \frac{4\pi }{3}\times (\frac{a}{2}{)}^2}{6\times \frac{1}{2}\times \frac{\sqrt{3}}{2}{a}^2\times \frac{\sqrt{8}}{3}a}$

$=\frac{\sqrt{2\pi }}{6}=74.05\%$

Hence, both option $B$ and $C$ are correct.