Calculate the packing efficiency in a simple cubic lattice. 

Explanation

a) Packing efficiency is a percentage of total space filled by the particles in a crystal.
The unit cell of a simple cube is

 A simple cubic unit cell consists of only one atom/ particle/ sphere per unit cell. Let .a. be the edge length of the unit cell and .r. be the radius of the sphere.
We know that volume of the sphere `=(4)/(3)pir^(3)`
The spheres/ lattice points at the corners in the simple cubic unit cell are touching each other. Therefore edge length `= a =2r`
Similarly volume of the cube `= a^(3) = 2r^(3) = 8r^(3)`
Percentage of the space occupied by spheres
`=("Volume of the sphere")/("Volume of the cube")xx100=((4)/(3)pir^(3))/(8r^(3))xx100`
`=(4xx100xx22)/(8xx3xx7)`
`=52.38%~~52.4%` is the packing fraction in simple cubic lattice