Question

calculate packing efficiency of a metal crystal for a simple lattice? ​

Answer 1

Packing efficiency in simple cubic laattice :

In a simple cubic lattice the atoms are located only on the corners of the cube

The particles touch each other along the edge.Thus the edge length or side of the cube 'a' and the radius of each particles,r are related as :

a=2ra=2r

The volume of the cubic unit cell =a3=(2r)3=8r3a3=(2r)3=8r3

Since a simple cubic unit cell contains only 1 atom.

The volume of the occupied space =4343πr3πr3

∴∴ Packing efficiency =Volume of one atomVolume of cubic unit cellVolume of one atomVolume of cubic unit cell×100×100

⇒43πr3×1008r3⇒43πr3×1008r3

⇒π6⇒π6×100×100

⇒52.4%

Answer 2

Hence a simple cubic lattice has atoms at their edges...........

Then the relation between the side of cube(a) and radius of atom(r) is

A=2r

The volume of simple cubic cell is

r³=(2r)³=8r³

Volume of an atom is

4/3πr³

Now

Packing efficiency= volume of an atom/volume of simple cubic cell×100

=>4/3πr³/8r³×100

=>4πr³/24r³×100

=>π/6×100

=>3.14/6×100 (π=3.14)

=>314/6=52.4%