Solid State - Result Question 20
####20. The packing efficiency of the two-dimensional square unit cell shown below is
(2010)
(a) $39.27 \text{\%}$
(b) $68.02 \text{\%}$
(c) $74.05 \text{\%}$
(d) $78.54 \text{\%}$
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Answer:
Correct Answer: 20. (d)
Solution:
- Contribution of circle from corner of square $=\frac{1}{4}$
$\Rightarrow$ Effective number of circle per square
$$ =\frac{1}{4} \times 4+1(\text { at centre })=2 $$
$\Rightarrow$ Area occupied by circle $=2 \pi r^{2}, r=$ radius.
Also, diagonal of square $4 r=\sqrt{2} L$, where $L=$ side of square.
$\Rightarrow$ Packing fraction $=\frac{\text { Area occupied by circles }}{\text { Area of square }}$
$$ =\frac{2 \pi r^{2}}{L^{2}}=\frac{2 \pi r^{2}}{8 r^{2}}=\frac{\pi}{4}=0.785 $$
$\Rightarrow \text{\%}$ packing efficiency $=78.5 \text{\%}$.
