Question

The packing efficiency of the two-dimensional square unit cell shown in the given figure is :

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

Answer 1

The correct option is D $78.54\%$

Let $a=2\sqrt{2}r$
Let us consider the square plane and $L$ be the length of square (given).
$\text{Area}=L^2$
By applying pythagoras theorem,
$L^2$ + $L^2$ = $(2r{)}^2$
$4r^2=$ $2L$
$\frac{\sqrt{2L}}{2}=r$
$\text{Area}=$ $2\pi \times {\frac{\sqrt{\left(2L\right)}}{4}}^2$
Packing efficiency $=\frac{2\times \pi r^2}{(2\sqrt{2}r{)}^2}=\frac{2\pi r^2}{8r^2}=\frac{\pi }{4}=78.5\%$