In a tetragonal crystal

Options

A. `a = b = c, alpha = beta = 90^(@) != gamma`
B. `alpha = beta = gamma = 90^(@), a = b != c`
C. `alpha = beta = gamma = 90^(@), a != b != c`
D. `alpha = beta = 90^(@), gamma = 120^(@), a = b != c`

Answer

To solve the question regarding the relationship between edge lengths and interfacial angles in a tetragonal crystal, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Tetragonal Crystal System**:
- In a tetragonal crystal system, there are two equal edge lengths and one different edge length. This means that the lengths of the edges can be represented as:
- \( a = b \neq c \)
- Here, \( a \) and \( b \) are the lengths of the edges in the base plane, while \( c \) is the height of the crystal.

2. **Identify the Interfacial Angles**:
- For the tetragonal system, the interfacial angles are defined as:
- \( \alpha = \beta = \gamma = 90^\circ \)
- This means that all angles between the edges are right angles.

3. **Summarize the Relationships**:
- From the above points, we can summarize the relationships as follows:
- Edge lengths: \( a = b \neq c \)
- Interfacial angles: \( \alpha = \beta = \gamma = 90^\circ \)

4. **Evaluate the Options**:
- Now, we can evaluate the given options based on our findings:
- **Option A**: Incorrect, as it states all edges are equal.
- **Option B**: Correct, as it states \( a = b \) and \( a \neq c \) with angles being \( 90^\circ \).
- **Option C**: Incorrect, as it suggests an incorrect relationship between edge lengths.
- **Option D**: Incorrect, as it states different interfacial angles.

5. **Conclusion**:
- The correct answer is **Option B**, which accurately reflects the relationships in a tetragonal crystal system.