Question

The packing efficiency for the 2-dimensional square unit cell shown below is

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

Answer 1

The correct option is D 78.54%

From the figure, we can say it is non-primitive and unit cell comprise of 2 atoms i.e., $Z=2$
In $△ABC$
$A{C}^2=A{B}^2+B{C}^2$
$(4r{)}^2=V^2+V^2$
$16r^2=2V^2$
$r^2=\frac{V^2}{8}$

$r=\frac{V}{\sqrt{8}}$
$Z_{eff}=2$
$\text{Area of square unit cell}=V\times V$
$\text{Packing Efficiency}=\frac{Z\times \text{Area of atom}}{\text{Area of unit cell}}\times 100$
$=\frac{Z\times \pi r^2}{V^2}\times 100$
$=\frac{2\times \pi V^2}{8V^2}\times 100$
$=\frac{\pi }{4}\times 100=78.54\%$