a) Calculate the packing efficiency of particles in a body centred cube. 

Explanation

Packing efficiency in BCC

The number of atoms per unit cell on bcc is two. Each atom considered as one sphere. Let edge length of the unit cell = a
Radius of sphere = r
Radius of sphere - c
Radius of face diagonal - b
In `DeltaEFD,FD^(2)=EF^(2)+ED^(2)`
`b^(2)=a^(2)+a^(2)`
`b^(2)=2a^(2)`
In `DeltaAFDAF^(2)=FD^(2)+AD^(2)`
`C^(2)=b^(2)+a^(2)`
`C^(2)=2a^(2)+a^(2)`
`C^(2)=3a^(2)`
`C=sqrt3a`
Since `C=4r`
`4r=sqrt3a`
`a=(4r)/(sqrt3)`
`"Volume of unit cell"=a^(3)=((4r)/(sqrt3))^(3)=(64r^(3))/(3sqrt3)`
Volume of two spheres `=2xx(4)/(3)pir^(3)=(8pir^(3))/(3)`
`="Packing efficiency"=("Volume occupied by 2 spheres")/("Total Volume unit cell")xx100%=((8pir^(3))/(3))/((64r^(3))/(3sqrt3))xx100%=68%`