Calculate packing efficiency in BCC lattice.

Explanation

From the figure it is clear that the atom at the centre will be in touch with other two atoms along the body diagonal.
Let a: be the edge length of the cube.
b: be the face diagonal
c: be the body diagonal.
r: isthe radius of the sphere and c=4r
In `triangleEFD`
`b^(2) =a^(2) + a^(2)`
`b^(2) =2a^(2)`
In `triangleAFD`
`c^(2) =a^(2) + b^(2)`
`c^(2) =a^(2) + 2a^(2) = 3a^(2)`
`c= sqrt(3) xx a`
`4r = sqrt(3) xx a` `(4r)/sqrt(3)`
Volume of the cube `=a^(3) = (4/sqrt(3)r)^(8)`
In this unit cell, total number of particles is 2 and volume is `2 xx 4/3 pi r^(3)`
Packing efficienty = `("Volume of two spheres in the unit cell")/("Total volume of the unit cell") xx 100`
`=(2 xx 4/3 pi r^(3))/[4/3 r]^(3) xx 100`
Packing efficiency = 68%