# Packing Efficiency Of A Unit Cell

A crystal lattice is made up of a very large number of unit cells where every lattice point is occupied by one constituent particle. The unit cell can be seen as a three dimension structure containing one or more atoms. We always observe some void spaces in the unit cell irrespective of the type of packing. Percentage of spaces filled by the particles in the unit cell is known as the packing fraction of the unit cell. In this section we shall learn about packing efficiency. Packing fraction of different types of packing in unit cells is calculated below:

### Packing efficiency in Hexagonal close packing and Cubic close packing structure:

Hexagonal close packing (hcp) and cubic close packing (ccp) have same packing efficiency. Let us take a unit cell of edge length “a”. Length of face diagonal, b can be calculated with the help of Pythagoras theorem,

Packing Efficiency of a Unit Cell

$b^2$= $a^2~ +~ a^2$

=> $b^2$ = $2a^2$

=> b = $\sqrt{2}$ a

From the figure, radius of the sphere, r = 4 × length of face diagonal, b

r = $\frac {d}{4}b$ = $\frac {\sqrt{2}}{4} a$

=> a = $2 \sqrt{2} r$

In ccp structures, each unit cell has four atoms,

Packing efficiency = $\frac{volume~ occupied~ by~ four~ spheres~ in~ unit~ cell}{Total~ volume~ of~ unit cell}$ × 100

= $\frac {4~×~\left( \frac 43 \right) \pi r^3~×~100}{( 2 \sqrt{2} r)^3}$ = 74%

### Packing efficiency in body centered cubic structure:

In body centered cubic unit cell, one atom is located at body center apart from corners of the cube. Let us take a unit cell of edge length “a”. Length of body diagonal, c can be calculated with help of Pythagoras theorem,

$c^2~=~ a^2~ + ~b^2$

Where b is the length of face diagonal, thus b = $\sqrt{2}~a$

=> $c^2~=~ 3a^2$

=> c = $\sqrt{3}$ a

From the figure, radius of the sphere, r = 4 × length of body diagonal, c

=> r = $\frac {c}{4}$ = $\frac{\sqrt{3}}{4}~a$

=> a = $\frac {4}{\sqrt{3}}$ r

In body centered cubic structures, each unit cell has two atoms,

Packing efficiency = $\frac{volume~ occupied~ by~ two~ spheres~ in~ unit~ cell}{Total~ volume~ of~ unit ~cell}$ × 100

= $\frac {2~×~\left( \frac 43 \right) \pi r^3~×~100}{( \frac {4}{\sqrt{3}})^3}$ = 68%

### Packing efficiency in simple cubic structure:

In simple cubic unit cell, atoms are located at the corners of cube. Let us take a unit cell of edge length “a”. Radius of atom can be given as,

r = $\frac a2$

=> a = 2r

In simple cubic structures, each unit cell has only one atom,

Packing efficiency = $\frac{volume~ occupied~ by~ one~ atom}{Total~ volume~ of~ unit ~cell}$ × 100

= $\frac {\left( \frac 43 \right) \pi r^3~×~100}{( 2 r)^3}$ = 52.4%

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