The packing efficiency of the two-dimensional | vedclass

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Question

(D) In the given two-dimensional square unit cell,there is one atom at the center and four atoms at the corners.The effective number of atoms $(n)$ is

Vedclass · Accepted Answer

$78.54$

Vedclass · Answer

(D) In the given two-dimensional square unit cell,there is one atom at the center and four atoms at the corners.The effective number of atoms $(n)$ is calculated as:$n = 1_{	ext{center}} + 4 	imes \frac{1}{4}_{	ext{corner}} = 1 + 1 = 2$The diagonal of the square $(AC)$ is equal to $4r$,where $r$ is the radius of the atom.$AC = 4r = \sqrt{2}a$,where $a$ is the edge length of the square.Therefore,$a = \frac{4r}{\sqrt{2}} = 2\sqrt{2}r$.The packing efficiency ($P$.$E$.) is the ratio of the area occupied by the atoms to the total area of the unit cell:$P.E. = \frac{n 	imes \pi r^2}{a^2} 	imes 100$$P.E. = \frac{2 	imes \pi r^2}{(2\sqrt{2}r)^2} 	imes 100 = \frac{2\pi r^2}{8r^2} 	imes 100 = \frac{\pi}{4} 	imes 100$$P.E. = \frac{3.14159}{4} 	imes 100 = 78.54\%$

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

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