if `a=b ne c` and `alpha=beta=gamma=90^(@)`, the crystal system is

Options

A. cubic
B. triclinic
C. hexagonal
D. tetragonal

Answer

To determine the crystal system based on the given conditions (a = b ≠ c and α = β = γ = 90°), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Parameters**:
- The parameters a, b, and c represent the lengths of the edges of the crystal unit cell.
- The angles α, β, and γ represent the angles between the edges.

2. **Analyze the Given Conditions**:
- We have a = b, which means two edge lengths are equal.
- We have c, which is not equal to a or b (c ≠ a, c ≠ b).
- The angles α, β, and γ are all equal to 90°. This indicates that the edges meet at right angles.

3. **Identify Possible Crystal Systems**:
- There are seven crystal systems: cubic, tetragonal, orthorhombic, hexagonal, rhombohedral, monoclinic, and triclinic.
- We need to check which crystal system matches the conditions given.

4. **Examine Each Crystal System**:
- **Cubic**: a = b = c and α = β = γ = 90° (not applicable here).
- **Tetragonal**: a = b ≠ c and α = β = γ = 90° (matches our conditions).
- **Orthorhombic**: a ≠ b ≠ c and α = β = γ = 90° (not applicable here).
- **Hexagonal**: a = b ≠ c, α = β = 90°, γ = 120° (not applicable here).
- **Monoclinic**: a ≠ b ≠ c and two angles are 90° (not applicable here).
- **Triclinic**: a ≠ b ≠ c and no angles are 90° (not applicable here).

5. **Conclusion**:
- The only crystal system that satisfies the conditions a = b ≠ c and α = β = γ = 90° is the **Tetragonal** crystal system.

### Final Answer:
The crystal system is **Tetragonal**.