Which of the following is the correct atomic packing factor (APF) for the hexagonal close-packed (hcp) crystal structure?

• A. 0.74
• B. 0.52
• C. 0.68
• D. 0.60

Answer: (A)

Verified step by step guidance

1. Understand that the Atomic Packing Factor (APF) is defined as the fraction of volume in a crystal structure that is occupied by atoms. It is calculated as the volume of atoms in a unit cell divided by the total volume of the unit cell.
2. Recall that the hexagonal close-packed (hcp) structure consists of 6 atoms per unit cell (counting the fractional contributions of atoms at corners and faces).
3. Calculate the volume occupied by atoms in the hcp unit cell using the formula for the volume of a sphere: \(V_{atom} = \frac{4}{3} \pi r^{3}\), and multiply by the number of atoms per unit cell.
4. Determine the volume of the hcp unit cell using the formula for the volume of a hexagonal prism: \(V_{cell} = \frac{3 \sqrt{3}}{2} a^{2} c\), where \(a\) is the edge length and \(c\) is the height of the unit cell. Note that in ideal hcp, \(c/a \approx 1.633\).
5. Calculate the APF by dividing the total atomic volume by the unit cell volume: \(APF = \frac{\text{number of atoms} \times V_{atom}}{V_{cell}}\). This calculation leads to the known APF value for hcp, which is approximately 0.74.