Question

A Metallic element crystallises 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?

Answer 1

For a hcp unit cell, there are 6 atoms per unit cell. If r is the radius of the metal atoms, volume occupied by the metallic-

Atoms $6\times \frac{4}{3}\times \pi r^3=6\times 1.33\times \frac{22}{7}\times r^3=25.08\times r^3$

Geometrically it has been shown that the base area of hcp unit cell

$=6\times \surd 3/4\times 4r^2$ and the height $=4r\times \surd 2/3$

Volume of the unit cell

$=Area\times height=6\times \surd 3/4\times 4r^2\times 4r\times \sqrt{2}/3=33.94r^3$

Volume of the empty space of one unit cell

$=33.94r^3–25.08r^3=8.86r^3$

$=\frac{8.816r^3}{33.94r^3}\times 100=26.1$%