Stracture of unit cell is described in the given figure packing fraction of the unit cell is?

Options

A. `0.521`
B. `0.907`
C. `0.093`
D. `0.745`

Answer

b

Explanation

` "Packing faction" = ("Area occupied by circles")/("area of unit cell")`
Let edge length `= l`
Thus, radius (r )` = (l)/(2)` from figure it is clear parallelegram has one complete circle
Thus area of circuit `= pi r^(2) = pi ((1)/(2))^(2)`
Area of parallelagram `= 2 (area of Delta ABC)`
Area of `Delta ABC = (1)/(2) BC xx AD`

`AD^(2) = AB^(2) - BD^(2)`
`= l^(2) - (l^(2))/(4) = (3l^(2))/(4)`
`AD = sqrt(3) ((1)/(2))`
Area of `Delta ABC = (1)/(2) ( 1 + sqrt(3) (1)/(2)) = = sqrt(3) (l^(2))/(4)`
Area of parallelogram = 2 + Area of `Delta ABC`
`= 2 sqrt(3) (l^(2))/(4) = sqrt(3) (l^(2))/(2)`
Packing fraction `(pi l 4)/(sqrt(3)l^(2)//2) = (pi)/(2 sqrt(3)) = 0.907`