Question:
The packing efficiency of the two-dimensional square unit cell shown below is
The packing efficiency of the two-dimensional square unit cell shown below is
Updated On: Apr 5, 2024
- $ 39.27 \% $
- $ 68.02 \% $
- $ 74.05 \% $
- $ 78.54 \% $
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The Correct Option is D
Solution and Explanation
Contribution of circle from comer 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{Area\, \, occupied\, \, by\, \, circles}{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 \% $
$ \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{Area\, \, occupied\, \, by\, \, circles}{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 \% $
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