Question

A metallic element crystallizes into a lattice containing a sequence of layers of ABABAB. Any packing of spheres leaves out voids in the lattice. What percentage by volume of this lattice is empty space?

• A. $26$
• B. $32$
• C. $48$
• D. $52$

Answer 1

The correct option is A $26$

As we know,

ABAB…. stacking pattern implies hexagonal closest packed structure.

Packing fraction in hcp $=\frac{6\times \frac{4}{3}\pi r^3}{\text{Volume}}$ $=\frac{6\times 4\pi r^3}{3\times \text{Area}\times \text{Height}}$

Area (of base) of unit cell in hcp $=6\sqrt{3}{r}^2$

Height of unit cell in hcp $=4r\times \frac{\sqrt{2}}{3}$

$\therefore$ Packing fraction $=\frac{6\times 4\pi r^3}{3\times 6\sqrt{3}{r}^2\times 4r\times \frac{\sqrt{2}}{3}}$ $=\frac{\pi }{3\sqrt{2}}=0.74or74\%$

Thus, empty space in hcp $=1-0.74=0.26$ or $26\%$.