Calculate the efficiency of packing in case of a metal crystal for
(i) simple cubic
(ii) body- centred cubic
(iii) face - centred cubic . (With the assumptions that atoms are touching each other).

Solution:

(i) Simple Cubic: A simple cubic unit cell has one sphere (or atom) per unit cell. If r is the radius of the sphere, then volume occupied by one sphere present in unit cell = $\frac{4}{3}{\mathrm{\pi r}}^3$
           
Edge length of unit cell (a) = r + r = 2r
Volume of cubic $\left({\mathrm{a}}^3\right) = (2\mathrm{r}{)}^3 = 8{\mathrm{r}}^3$
Volume of occupied by sphere  = $\frac{4}{3}{\mathrm{\pi r}}^3$
Percentage volume occupied = percentage of efficiency of packing

$$\begin{gathered} = \frac{\mathrm{Volume} \mathrm{of} \mathrm{sphere}}{\mathrm{Volume} \mathrm{of} \mathrm{cube} }\times 100 \\ = \frac{{\displaystyle \frac{4}{3}}{\mathrm{\pi r}}^3}{8{\mathrm{r}}^3}\times 100 \\ = \frac{1}{6}\times 3.143\times 100 = 52.4\% \end{gathered}$$

For simple cubic metal crystal the efficiency of packing = 52.4%.

(b) Body centred cubic: From the figure it is clear that the atom at the centre will be in touch with other two atoms diagonally arranged and shown with solid boundaries.

$$\begin{gathered} \mathrm{In} ∆\mathrm{EFD} {\mathrm{b}}^2 = {\mathrm{a}}^2+{\mathrm{a}}^2 \\ = 2{\mathrm{a}}^2 \\ \mathrm{b} = \sqrt{2\mathrm{a}} \\ \mathrm{Now} \mathrm{in} ∆\mathrm{AFD} \\ {\mathrm{c}}^2 = {\mathrm{a}}^2+ {\mathrm{b}}^2 \\ = {\mathrm{a}}^2+2{\mathrm{a}}^2 = 3{\mathrm{a}}^2 \\ \mathrm{c} = \sqrt{3\mathrm{a}} \end{gathered}$$

The length of the body diagonal c is equal to 4r where r is the radius of the sphere (atom). But c = 4r, as all the three spheres along the diagonal touch each other

∴              $\sqrt{3\mathrm{a}} = 4\mathrm{r}$

or                   $\mathrm{a}=\frac{4\mathrm{r}}{\sqrt{3}}$

or                    $\mathrm{r}=\frac{\sqrt{3}}{4}\mathrm{a}.$



Fig. BCC unit cell. In this sort of structure total number of atoms is two and their volume is $2\times \left(\frac{4}{3}\right){\mathrm{\pi r}}^3.$

Volume of the cube, a³ will be equal to
$\left(\frac{4}{\sqrt{3}}\mathrm{r}\right) \mathrm{or} {\mathrm{a}}^3 = {\left(\frac{4}{\sqrt{3}}\mathrm{r}\right)}^3.$

Therefore, Percentage of efficiency Volume occupied by four - spheres

$$\begin{gathered} = \frac{\mathrm{in} \mathrm{the} \mathrm{unit} \mathrm{cell}}{\mathrm{Total} \mathrm{volume} \mathrm{of} \mathrm{the} \mathrm{unit} \mathrm{cell}}\times 100\% \\ = \frac{2\times \left({\displaystyle \frac{4}{3}}\right){\mathrm{\pi r}}^3\times 100}{(4/\sqrt{3}\mathrm{r}{)}^3}\% \\ = \frac{(8/3) {\mathrm{\pi r}}^3 \times 100}{[64/(3\sqrt{3}\left){\mathrm{r}}^3\right]}\% = 68\% \end{gathered}$$

(c) Face centred cubic: A face centred cubic cell (fcc) contains four spheres (or atoms) per unit volume occupied by one sphere of radius

r = 4/3 $\mathrm{\pi }$r³

Volume occupied by four spheres present in the unit cell

r = 4/3 $\mathrm{\pi }$r³ x 4 = 16/3 $\mathrm{\pi }$r³





Fig.  Face centred cubic unit cell. Figure indicates that spheres placed at the corners touches a face centred sphere. Length of the face diagonal
= r + 2r + r = 4r

Hence, for face centred cubic, efficiency of packing = 74%.