The packing efficiency of the two dimensional square unit cell show in the adjoining fig ,. Is
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Options

A. 0.3927
B. 0.6802
C. 0.7405
D. 0.7854

Answer

D

Explanation

In the two - dimensional packing , packing fraction
Area occupied by circles
` (" within the square")/("Area of the square") `
` ( 2 xx pir^(2))/a^(2) `
AC = `sqrt(AB^(2) + BC^(2)) = sqrt(a^(2) + a^(2)) = sqr2 a `
but AC = 4 r( r= radius of each sphere)
` sqrt2 a= 4r or a = 4/sqrt2 r = 2 sqrt2r`
packing fraction
` ( 2xx pi r^(2))/((2 sqrt2 r)^(2))= pi/4 = ( 3.143)/4 = 0.785 i.e, 78.5%`