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

Explanation

The number of atoms per unit cell in bcc structure is two. Each atom is considered as one sphere.
Let the edge length of the unit cell = a
Radius of the sphere = r
Length of the body diagonal = c
Length of the body diagonal = b
In `DeltaEFD`
`FD^(2)=EF^(2)+DE^(2)`
`b^(2)=a^(2)+a^(2)=2a^(2)`
Now in `DeltaAFD`
`AF^(2)=AD^(2)+FD^(2)`
Now in `DeltaAFD`
`AF^(2)=AD^(2)+FD^(2)`
`c^(2)=a^(2)+b^(2)`
`c^(2)=a^(2)+2a^(2)=3a^(2)`
`c =sqrt3a`
`W.K.T c=4r`
Therefore `=4r=sqrt3a`
`a=(4r)/(sqrt3)`

`"Total volume of the unit cell "= a^(3) = ((4r)/(sqrt3))^(3)`
Since bcc lattice contains 2 atoms per unit cell
`"Volume of two spheres "=2 xx(4)/(3)pir^(3)`
`"Packing efficiency "= ("Volume of the two spheres in unit cell")/("Total volume of the unit cell")xx100%`
`=(2xx(4)/(3)pir^(3))/(((4r)/(sqrt3))^(3))xx100%`
`=((8)/(3)pir^(3))/((64)/(3sqrt3)r^(3))xx100=68%`