The packing efficiency of the two-dimensional | vedclass

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(D) In the given two-dimensional square unit cell,the atoms are present at the four corners and one at the center of the square.Let the side length of

Vedclass · Accepted Answer

$78.54$

Vedclass · Answer

(D) In the given two-dimensional square unit cell,the atoms are present at the four corners and one at the center of the square.Let the side length of the square be $a$ and the radius of each atom be $r$.The diagonal of the square is $d = a \sqrt{2}$.Along the diagonal,the atoms are in contact,so $d = 4r$.Therefore,$a \sqrt{2} = 4r$,which gives $a = 2 \sqrt{2} r$.The number of atoms per unit cell is $Z = 4 	imes (1/4) + 1 = 2$.The area of the unit cell is $a^2 = (2 \sqrt{2} r)^2 = 8r^2$.The area occupied by the atoms is $Z 	imes \pi r^2 = 2 \pi r^2$.Packing efficiency = $\frac{	ext{Area occupied by atoms}}{	ext{Total area of unit cell}} 	imes 100$Packing efficiency = $\frac{2 \pi r^2}{8 r^2} 	imes 100 = \frac{\pi}{4} 	imes 100 \approx 0.7854 	imes 100 = 78.54 \%$.

The packing efficiency of the two-dimensional square unit cell shown below is (in $\%$)

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