Question

Calculate packing efficiency in simple cubic lattice .

Answer 1

The length or side of the cube be ‘$a$’, and the radius of each particle be ‘$r$’. $a=2r$ Volume of cube $=a^3$

$$\begin{gathered} ={\left(2r\right)}^3 \\ =8r^3 \end{gathered}$$

In the simple cube, each unit cell has only one sphere, The volume of sphere $=\frac{4}{3}{\mathrm{\pi r}}^3$

Packing efficiency is given by $r$

Packing efficiency $=\frac{\mathrm{volume}\mathrm{of}\mathrm{sphere}}{\mathrm{volume}\mathrm{of}\mathrm{cube}}\times 100$

$$\begin{gathered} =\frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}{8r^3}\times 100 \\ =52.4\% \end{gathered}$$

Hence , the required value is $52.4\%$ .