Question

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

• A. $39.27\%$
• B. $68.02\%$
• C. $74.05\%$
• D. $78.57\%$

Answer 1

Answer: (D)

$78.57\%$

Here the diagonal is 4r, where, r is the radius of atom

We know diagonal $=\sqrt{(}{L}^2+L^2)=\sqrt{2\times L}$

$4\times r=\sqrt{2}\times L$

$L=\frac{4\times r}{\sqrt{2}}$

Number of spheres inside the square are $1+4\times \frac{1}{4}$

Area of sphere $=\pi \times r\times r$

Packing fraction $=\frac{2\times \pi \times r\times r}{8\times r^2}=\frac{\pi }{4}=0.7857$

Percentage is $78.57\%$