Question

For the hexagonal close-packed crystal: (12 points) a. How many atoms are associated with the unit cell? b. What is the coordination number? c. What is the atomic packing factor?

Answer 1

Answer :

(a) Total number of atoms in the cell = 6

(b) The coordination number of HCP is, 12

(c) The atomic packing factor is, 74.0 %

Explanation :

Hexagonal close-packed structure (HCP) :

Crystal structure : The manner in which the atoms or ions are spatially arranged.

(a) The HCP unit cell consist of three primitive unit cells in a hexagonal prism. That means, 3 atoms are in the middle layer and 12 atoms are at top and bottom layers and 2 face atoms.

So, the total number of atoms in the cell = 3 + (1/2)×2 + (1/6)×6 × 2 = 6

(b) Coordination number : It is defined as any constituent particle in a crystal is the number of its nearest neighbours.

The coordination number of HCP is, 12

(c) Packing fraction or factor : It is defined as the percentage of the total space filled by the particles.

$\text{Packing factor}=\frac{n\times V_{atom}}{V_{cell}}\times 100$

$\text{Packing factor}=\frac{6\times (\frac{4}{3}\pi r^3)}{24\sqrt{2}r^3}\times 100$

$\text{Packing factor}=74.0\%$