Question

Packing density in identical solid spheres is $74\%$ in:

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

Answer 1

Answer: (B), (C)

The correct options are
B hcp
C fcc

For fcc:

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

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

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

For hcp:

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

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

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