A bcc lattice is made up of hollow spheres of `B`. Spheres of solids `A` are present in hollow spheres of `B`. The radius of `A` is half of the radius of `B`. The ratio of total volume of spheres of `B` unoccupied by `A` in a unit cell and volume of unit cell is `A xx (pisqrt(3))/(64)`. Find the value of `A`.

Answer

7

Explanation

Let radius of hollow sphere B=r
`therefore` Edge length (a) `=4r//sqrt(3)`
Volume of unit cell `=a^(3)=(4r//sqrt(3))^(2)`
Volume of B unoccupied by A (having radius=r/2)
in unit cell `=2xx[(4)/(3)pir^(3)-(4)/(3)pi((r)/(2))^(3)]`
Volume of B unoccupied by A
`therefore ("in unit cell")/("volume of unit cell")=`((4)/(3)pixx(7r^(3))/(3)xx2)/(((4r)/sqrt(3))^(3))=(7pisqrt(3))/(64)`
`therefore Axx(7pisqrt(3))/(64)=(7pisqrt(3))/(64)`
`therefore A=7`