Question

The coordination number and volume of unit cell of hexagonal closed packed structure are respectively:

• A. 6 and \( \left(\frac{3\sqrt{3}}{4}a^2\right)c^2 \)
• B. 8 and \( \left(\frac{4\sqrt{3}}{5}a^2\right)c^3 \)
• C. 10 and \( \left(\frac{4\sqrt{3}}{5}a^2\right)c^2 \)
• D. 12 and \( \left(\frac{3\sqrt{3}}{2}a^2\right)c \)

Hint:

For close-packed structures like HCP and FCC, the coordination number is always 12, which is the maximum possible for spheres of equal size. This can help you quickly eliminate options with incorrect coordination numbers.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for two fundamental properties of the Hexagonal Close-Packed (HCP) crystal structure: its coordination number and the volume of its conventional unit cell.
Step 2: Detailed Explanation:
Coordination Number (CN):
The coordination number is the number of nearest neighbors to any given atom. In the HCP structure:

An atom has 6 nearest neighbors in its own hexagonal layer.
It has 3 nearest neighbors in the layer directly above it.
It has 3 nearest neighbors in the layer directly below it.
Therefore, the total coordination number is \( 6 + 3 + 3 = 12 \).
Volume of the Unit Cell (V):
The conventional unit cell of an HCP structure is a prism with a hexagonal base and height 'c'.

The base is a regular hexagon with side length 'a'.
The area of a regular hexagon can be calculated as the area of six equilateral triangles with side 'a'.
Area of one equilateral triangle = \( \frac{\sqrt{3}}{4}a^2 \).
Area of the hexagonal base = \( 6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2 \).
The volume of the unit cell is the base area multiplied by the height 'c'.
\( V = (\text{Base Area}) \times (\text{height}) = \left(\frac{3\sqrt{3}}{2}a^2\right)c \).

Step 3: Final Answer:
The coordination number for HCP is 12, and the volume of the unit cell is \( \frac{3\sqrt{3}}{2}a^2c \). This matches option (D).