Calculate the packing efficiency of the two-dimensional square unit cells as given in the figure.

Explanation

Each corner sphere of the unit cell is equally shared by 4 adjoining unit cells. So, each corner sphere makes a contribution of 1/4 to the unit cell. The sphere at the centre of the unit cell is not shared by other unit cells, so its contribution to the unit cell is 1. Number of corners of the unit cell is 4.

Total number of particles per unit cell = `4xx1/4+1=2`
Area of each sphere is `pir^(2)`, where r = radius of the sphere.
Total area for two particles in the unit cell = `2pir^(2)`
The area of the unit cell = `a^(2)`. From the figure it is evident,
`AB=sqrt(AC^(2)+CB^(2))=sqrt(a^(2)+a^(2))=asqrt2=4r`
`therefore4r=asqrt2or,a=2sqrt2r`
Hence, packing fraction = `("Total area of 2 spheres")/("Area of the unit cell")xx100`
`=(2pir^(2))/a^(2)xx100=(2pir^(2))/((2sqrt2r)^(2))=78.5%`