The crystal system of a compound with unit cell dimensions a = 0.388 , b = 0.388 and c = 0.506 nm and `alpha = beta = 90^@ and gamma = 120^@` is

Options

A. Hexagonal
B. Cubic
C. Rhombohedral
D. Orthorhombic

Answer

To determine the crystal system of the compound with the given unit cell dimensions and angles, we will analyze the provided data step by step.

### Given Data:
- Unit cell dimensions:
- \( a = 0.388 \, \text{nm} \)
- \( b = 0.388 \, \text{nm} \)
- \( c = 0.506 \, \text{nm} \)
- Angles:
- \( \alpha = 90^\circ \)
- \( \beta = 90^\circ \)
- \( \gamma = 120^\circ \)

### Step 1: Identify the equality of the axes
- We observe that \( a = b \) and \( c \) is not equal to \( a \) or \( b \). This indicates that the lengths of the axes are not all equal.

**Hint:** Check if any two of the axes are equal and if the third is different.

### Step 2: Analyze the angles
- The angles \( \alpha \) and \( \beta \) are both \( 90^\circ \), while \( \gamma \) is \( 120^\circ \). This is a key characteristic of certain crystal systems.

**Hint:** Remember the significance of the angles in defining crystal systems.

### Step 3: Compare with known crystal systems
1. **Hexagonal:**
- Conditions: \( a = b \neq c \), \( \alpha = \beta = 90^\circ \), \( \gamma = 120^\circ \)
- This matches our data.

2. **Cubic:**
- Conditions: \( a = b = c \), \( \alpha = \beta = \gamma = 90^\circ \)
- This does not match.

3. **Rhombohedral:**
- Conditions: \( a = b = c \), \( \alpha = \beta = \gamma \) (but not \( 90^\circ \))
- This does not match.

4. **Orthorhombic:**
- Conditions: \( a \neq b \neq c \), \( \alpha = \beta = \gamma = 90^\circ \)
- This does not match.

**Hint:** Compare the conditions of the crystal systems with your observations about the axes and angles.

### Step 4: Conclusion
Since the conditions for the hexagonal system are satisfied (i.e., \( a = b \neq c \) and \( \alpha = \beta = 90^\circ \), \( \gamma = 120^\circ \)), we can conclude that the crystal system of the compound is **hexagonal**.

### Final Answer:
The crystal system of the compound is **hexagonal**.

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