What are the crystallographic parameter of hexagonal, monoclinic and triclinic unit cell respectively

Answer

To determine the crystallographic parameters of hexagonal, monoclinic, and triclinic unit cells, we need to understand the definitions and characteristics of each type of unit cell. The crystallographic parameters include the lengths of the edges (denoted as a, b, c) and the angles between them (denoted as α, β, γ).

### Step-by-Step Solution:

1. **Hexagonal Unit Cell:**
- **Edge Lengths:**
- a = b (the lengths of the edges in the base plane are equal)
- c (the height of the cell) is different from a and b.
- **Angles:**
- α = 90° (the angle between edge b and c)
- β = 90° (the angle between edge a and c)
- γ = 120° (the angle between edge a and b)

Thus, the parameters for the hexagonal unit cell are:
- a = b ≠ c
- α = β = 90°, γ = 120°

2. **Monoclinic Unit Cell:**
- **Edge Lengths:**
- a ≠ b ≠ c (all edges are of different lengths)
- **Angles:**
- α = γ = 90° (the angles between edges a and c, and b and c are right angles)
- β (the angle between edges a and b) is not equal to 90°.

Thus, the parameters for the monoclinic unit cell are:
- a ≠ b ≠ c
- α = γ = 90°, β ≠ 90°

3. **Triclinic Unit Cell:**
- **Edge Lengths:**
- a ≠ b ≠ c (all edges are of different lengths)
- **Angles:**
- α ≠ β ≠ γ (none of the angles are equal to each other and none are right angles).

Thus, the parameters for the triclinic unit cell are:
- a ≠ b ≠ c
- α ≠ β ≠ γ

### Summary of Crystallographic Parameters:
- **Hexagonal:**
- a = b ≠ c
- α = β = 90°, γ = 120°

- **Monoclinic:**
- a ≠ b ≠ c
- α = γ = 90°, β ≠ 90°

- **Triclinic:**
- a ≠ b ≠ c
- α ≠ β ≠ γ