Question

Show that the packing fraction in a simple cubic unit cell is 0.524 and the radius of the atom is equal to half of the edge length.

Answer 1

A simple cubic unit cell contains only one atom. Suppose the edge length of a simple cubic unit cell is a, and the atom’s radius is r.

In a simple cubic unit cell, the two spheres at the cube’s corners touch, as seen in Fig.

a = 2r   

Number of atoms per unit cell = `8 xx 1/8 = 1`

Volume of an atom = `4/3 pi r^3`

Volume of unit cell (cube) = a³

Volume of unit cell (cube) = (2)³

Volume of unit cell (cube) = 8r³

∴ Packing fraction = `"Volume occupied by atoms"/"Volume of unit cell"`

= `((4/3) pi r^3)/(8 r^3)`

= 0.524