Question

Calculate the efficiency of packing in case of a metal crystal for the following crystal structure (with the assumptions that atoms are touching each other):

(i) simple cubic

(ii) body-centered cubic

(iii) face-centered cubic

Answer 1

(i) The efficiency of packing in case of simple cubic unit cell is given below:
A simple cubic unit cell contains one atom per unit cell.
Also, $a=2r$, where $a$ is the edge length and $r$ is the radius of atom.
Total volume of unit cell $=a^3$.
Packing efficiency $=\frac{\text{Volume of one sphere}}{\text{Total volume of unit cell}}\times 100$

Packing efficiency $=\frac{\frac{4}{3}\pi r^3}{8r^3}\times 100=52.4\%$
(ii) The efficiency of packing in case of body-centred cubic unit cell is given below:
A body-centred cubic unit cell contains two atoms per unit cell.
Also, $\sqrt{3}a=4r$, where $a$ is the edge length and $r$ is the radius of atom.
Total volume of unit cell $=a^3$.
Packing efficiency $=\frac{\text{Volume of two spheres}}{\text{Total volume of unit cell}}\times 100$

Packing efficiency $=\frac{\frac{8}{3}\pi r^3}{\frac{64}{3\sqrt{3}}\times r^3}\times 100=68\%$
(iii) The efficiency of packing in case of face-centered cubic unit cell (with the assumptions that atoms are touching each other) is given below:
A face-centered cubic unit cell contains four atoms per unit cell.
Also, $a=2\sqrt{2}r$, where $a$ is the edge length and $r$ is the radius of atom.
Total volume of unit cell $=a^3$

Packing efficiency $=\frac{\text{Volume of four spheres}}{\text{Total volume of unit cell}}\times 100$

Packing efficiency $=\frac{\frac{16}{3}\pi r^3}{16\sqrt{2}{r}^3}\times 100=74\%$