The packing fraction of a simple cubic crystal with an interstitial atom exactly fitting at the body center is :

Explanation

Packing efficiency in body centred cubic lattice. If a is the edge length of the cube,
Body diagonal of the cube = `sqrt(3)a`
In bcc lattice atoms touch each other along the body diagonal of the cube.
`thereforesqrt(3)a=4rora=(4r)/(sqrt(3))`
In bcc structure, number of atoms per unit cell = 2
`therefore"Packing efficiency"=(2xx"Volume of one atom")/("Volume of unit cell")xx100`
`=(2xx4/(3)pir^(3))/(a^(3))xx100=(8/(3)pir^(3))/(((4pi)/(sqrt(3)))^(3))xx100`
`=8/(3)xx(3sqrt(3))/(64)(pir^(3))/(r^(3))xx100=(sqrt(3)xxpixx100)/(8)=(25sqrt(3)pi)/(2)=68%` (approx.)