(i) In a simple cubic unit cell :Suppose the edge length of the unit cell = a and radius of the sphere = r As spheres are touching each other, evidently, $a = 2r$No. of spheres per unit cell $= \frac{1}{8} \times 8 = 1$Volume of the sphere $= \frac{4}{3} \pi r^{3}$Volume of the cube $= a^{3} = (2r)^{3} = 8 r^{3}$ $\text{therefore}$ Fraction occupied. $\text{, i.e.,}$ packing fraction $= \left( \frac{4}{3} \pi r^{3} \right)$ / $8 r^{3} = 0.524$or % occupied $\text{i.e.,}$ packing efficiency = 52.4%(ii) In face-centred cubic structure : As sphere on the face-centre is touching the spheres at the corners, evidently $AC = 4r.$But from right angled triangle ABC, $AC = \sqrt{AB^{2} + BC^{2}}$ $\;\;\;\;= \sqrt{a^{2}+a^{2}} = \sqrt{2} a$ $\therefore\;\;\sqrt{2a = 4r}$ or $a = \frac{4}{\sqrt{2}} r$ $\therefore\;\;$ Volume of the unit cell $= a^{3} = \left( \frac{4}{\sqrt{2}} r \right)^{3} = \frac{32}{\sqrt{2}} r^{3}$No. of spheres in the unit cell $= 8 \times \frac{1}{8} + 6 \times \frac{1}{2} = 4$Volume of four spheres $= 4 \times \frac{4}{3} \pi r^{3} = \frac{16}{3} \pi r^{3}$∴ Fraction occupied i.e., packing fraction $= \frac{ \frac{16 \pi r^{3}}{3} }{ \frac{32 r^{3}}{\sqrt{2}} } = 0.74$or % occupied $\text{i.e.,}$ packing efficiency = 74%(iii) In body-centred cubic structure : As the sphere at the body-centre touches the spheres at the corners, body diagonal, $AD = 4r.$Further, face diagonal, $AC = \sqrt{AB^{2} + BC^{2}} = \sqrt{a^{2} + a^{2}} = \sqrt{2} a$and body diagonal, $AD = \sqrt{AC^{2} + CD^{2}} = \sqrt{2a^{2} + a^{2}} = \sqrt{3} a$ $\therefore\;\;\;\sqrt{3} a = 4 r$ or $a = \frac{4r}{\sqrt{3}}$ $\therefore\;\;\;$ Volume of the unit cell $= a^{3} = \left( \frac{4r}{\sqrt{3}} \right)^{3} = \frac{64 r^{3}}{3 \sqrt{3}}$No. of spheres per unit cell $= 8 \times \frac{1}{8} + 1 = 2$Volume of two spheres $= 2 \times \frac{4}{3} \pi r^{3} = \frac{8}{3} \pi r^{3}$$\text{therefore}$ Fraction occupied i.e., packing fraction $= \frac{ \frac{8}{3} \pi r^{3} }{ \frac{64 r^{3}}{3 \sqrt{3}} } = 0.68$or % occupied $\text{i.e}$ ., packing efficiency = 68%