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. Orthohombic

Answer

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

### Step 1: Identify the Unit Cell Dimensions
The unit cell dimensions are given as:
- \( a = 0.388 \, \text{nm} \)
- \( b = 0.388 \, \text{nm} \)
- \( c = 0.506 \, \text{nm} \)

### Step 2: Identify the Angles
The angles are given as:
- \( \alpha = 90^\circ \)
- \( \beta = 90^\circ \)
- \( \gamma = 120^\circ \)

### Step 3: Compare the Edge Lengths
From the dimensions:
- \( a = b \) (both are equal to 0.388 nm)
- \( c \) is not equal to \( a \) or \( b \) (0.506 nm)

### Step 4: Compare the Angles
From the angles:
- \( \alpha = \beta = 90^\circ \) (both are equal)
- \( \gamma = 120^\circ \) (not equal to \( \alpha \) and \( \beta \))

### Step 5: Determine the Crystal System
Now, we can classify the crystal system based on the relationships between the edge lengths and angles:
1. **Hexagonal System**:
- Conditions: \( a = b \neq c \) and \( \alpha = \beta = 90^\circ \), \( \gamma = 120^\circ \)
2. **Tetragonal System**:
- Conditions: \( a = b = c \) and \( \alpha = \beta = \gamma = 90^\circ \)
3. **Cubic System**:
- Conditions: \( a = b = c \) and \( \alpha = \beta = \gamma = 90^\circ \)
4. **Rhombohedral System**:
- Conditions: \( a = b = c \) and \( \alpha = \beta = \gamma \neq 90^\circ \)
5. **Orthorhombic System**:
- Conditions: \( a \neq b \neq c \) and \( \alpha = \beta = \gamma = 90^\circ \)

### Conclusion
Since we have \( a = b \neq c \) and \( \alpha = \beta = 90^\circ \) and \( \gamma = 120^\circ \), the crystal system of the compound is **Hexagonal**.