Question

Which of the following lattices has the highest packing efficiency :
A) Simple cubic
B) Body-centred cubic
C) Hexagonal close-packed lattice

Answer 1

Packing efficiency of Simple cubic unit cell
Here, atoms touch each other along edges.
Let $r=$ radius of atom and $a=$ edge length, then,

$d=a$ or $r=\frac{a}{2}$
Since a simple cubic unit cell contains only 1 lattice points.

$\therefore$ Volume of the occupied space $=\frac{4}{3}\times \pi r^3$
We know,

Packing efficiency

$=\frac{\text{Volume of occupied space}}{\text{Total volume of the unit cell}}\times 100\%$

Therefore,
Packing efficiency$=\frac{\frac{4}{3}\pi r^3}{(2r{)}^3}\times 100\%$
$=52.4\%$

Packing efficiency of Body centred cubic unit cell

Here, atoms touch each other along the body diagonal.
The length of the body diagonal

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

In this type of structure, total number of lattice points is 2 and their volume is

$2\times \frac{4}{3}\pi r^3$
$PF=\frac{\text{Volume of occupied space}}{\text{Total volume of the unic cell}}\times 100\%$

Therefore, $PF=\frac{2\times \left(\frac{4}{3}\right)\pi r^2\times 100}{{\left[\left(\frac{4}{\sqrt{3}}\right)r\right]}^3}\%$
$=68\%$

Packing efficiency of hexagonal close-packed lattice

Effective number of atoms in hcp unit cell are $6$.

We know,

$PF=\frac{\text{Volume of occupied space}}{\text{Total volume of the unic cell}}\times 100\%$

$PF=\frac{\text{Volume of occupied space}}{\text{Area of the base}\times \text{Height of unit cell}}\times 100\%$

Therefore, $PF=\frac{6\times \frac{4}{3}\pi r^3}{6\times \frac{\sqrt{3}}{4}(2r{)}^3\times 4r\sqrt{\frac{2}{3}}}\times 100\%$
$=74\%$

Comparison of packing efficiency
The packing efficiency in lattice is as follows:

• Simple cubic $=52.4\%$
• BCC $=68\%$
• HCP $=74\%$

Hence the maximum packing efficiency is of hexagonal packing lattice.

Final answer: Hexagonal close-packed lattice has highest packing efficiency.