Calculate the percentage efficiency of packing in case of body centered cubic crystal.

Explanation

Packing efficiency : in body centered cubic arrangement the spheres are touching along the leading diagonal of the cube as shown in the figure.
`AC^(2) - AB^(2) + BC^(2)`
`AC = sqrt(a^(2) +a^(2)) = sqrt(2a^(2) )= sqrt(2a)`
In `Delta ACG`
`AG^(2) = AC^(2) +CG^(2)`
`AG = sqrt((2a)^(2) +a^(2))`
i.e., `sqrt(3) a= 4r`
Volume of the sphere with radius r
Number of sphere belong to a unit cell in bcc arrangement is equal to two and bence the total volume of all spheres

Packin fraction = `("Total volume occupied by spheres in a unit cell ") /("Volume of the unit cell") xx 100`
Packing fraction `=((sqrt(3) pi a^(3))/( 8))/((a^(3))) xx 100`
`=1.732 xx 3.14 xx 12.5 =68%`