Question

The packing efficiency of the two-dimensional square unit cell shown below is

• A. $39.27\%$
• B. $68.02\%$
• C. $74.05\%$
• D. $78.54\%$

Answer 1

Answer: (D)

Contribution of circle from corner of square $=\frac{1}{4}$
$\Rightarrow$ Effective number of circle per square
$=\frac{1}{4}\times 4+1$ (at centre) $=2$
$\Rightarrow$ Area occupied by circle $=2\pi r^2,r=$ radius.
Also, diagonal of square $4r=\sqrt{2}L$, where $L=$ side of square.
$\Rightarrow$ Packing fraction $=\frac{\text{ Area occupied by circles }}{\text{ Area of square }}$
$=\frac{2\pi r^2}{L^2}=\frac{2\pi r^2}{8r^2}=\frac{\pi }{4}=0.785$
$\Rightarrow \%$ packing efficiency $=78.5\%$.