Calculate the packing efficiency in a simple cubic lattice. 

Explanation

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