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: Problem Definition:
The problem requires identifying the coordination number and unit cell volume for the Hexagonal Close-Packed (HCP) crystal structure.
Step 2: Solution:
Coordination Number (CN):
The coordination number represents the number of nearest neighbors surrounding an atom. For HCP:

Each atom has 6 neighbors within its hexagonal plane.
It has 3 neighbors in the layer above.
It has 3 neighbors in the layer below.
Therefore, CN = \( 6 + 3 + 3 = 12 \).
Unit Cell Volume (V):
The HCP unit cell is a hexagonal prism with base side 'a' and height 'c'.

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

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