Which of the following sets of axial angles and axial lengths represent maximum number in Bravais lattices?

Options

A. `alpha = beta = gamma = 90° and a = b ne c`
B. `alpha = beta = gamma = 90° and a ne b ne c`
C. `alpha = beta = gamma ne 90° and a = b = c`
D. `alpha = beta = gamma = 90° and a = b = c`

Answer

To solve the question regarding which set of axial angles and axial lengths represents the maximum number of Bravais lattices, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Bravais Lattices**:
- Bravais lattices are distinct arrangements of points in space. In three-dimensional space, there are 14 unique Bravais lattices.

2. **Identifying Axial Angles and Lengths**:
- The axial angles are denoted as α, β, and γ, and the axial lengths are A, B, and C.
- The relationships between these angles and lengths help classify the type of lattice.

3. **Analyzing Each Option**:
- **Option A**: α = β = γ = 90° and A = B ≠ C (Tetragonal)
- Tetragonal lattices correspond to 2 Bravais lattices.
- **Option B**: α = β = γ = 90° and A ≠ B ≠ C (Orthorhombic)
- Orthorhombic lattices correspond to 4 Bravais lattices.
- **Option C**: α = β = γ ≠ 90° and A = B = C (Trigonal)
- Trigonal lattices correspond to 1 Bravais lattice.
- **Option D**: α = β = γ = 90° and A = B = C (Cubic)
- Cubic lattices correspond to 3 Bravais lattices.

4. **Comparing the Number of Bravais Lattices**:
- From the analysis:
- Tetragonal: 2 Bravais lattices
- Orthorhombic: 4 Bravais lattices
- Trigonal: 1 Bravais lattice
- Cubic: 3 Bravais lattices
- The orthorhombic lattice has the maximum number of Bravais lattices, which is 4.

5. **Conclusion**:
- The option that represents the maximum number of Bravais lattices is **Option B** (Orthorhombic).

### Final Answer:
**Option B** (α = β = γ = 90° and A ≠ B ≠ C) represents the maximum number of Bravais lattices.

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