Question

Which of the three-dimensional primitive lattices has a unit cell where none of the internal angles is \(90^{\circ}\) ? (a) Orthorhombic, (b) hexagonal, (c) rhombohedral, (d) triclinic, (e) both rhombohedral and triclinic.

Short answer

Based on the analysis of the given lattice types, the short answer to the question is (e) both rhombohedral and triclinic, as these are the two three-dimensional primitive lattices where none of the internal angles is 90 degrees.

Step-by-step solution

Recall Characteristics of Given Lattice Types

Here's a basic rundown of each lattice type and their internal angles: (a) Orthorhombic: Internal angles are α = β = γ = 90°. (b) Hexagonal: Internal angles are α = β = 90°, and γ = 120°. (c) Rhombohedral: Internal angles are α = β = γ ≠ 90°. (d) Triclinic: Internal angles are α ≠ β ≠ γ ≠ 90°. Now that we have the characteristics of each lattice type, we can identify the lattice(s) that fit the criteria of having no internal angles equal to 90 degrees.

Identify Lattices without 90° Internal Angles

We will go through each options and evaluate whether any of the internal angles is 90 degree or not. (a) Orthorhombic: All internal angles are 90°, so this does not meet the criteria. (b) Hexagonal: Two internal angles are 90°, so this does not meet the criteria. (c) Rhombohedral: No internal angles are 90°, so this meets the criteria. (d) Triclinic: No internal angles are 90°, so this meets the criteria.

Conclusion

Considering the conditions provided in the problem, the answer is (e) both rhombohedral and triclinic. These are the two three-dimensional primitive lattices with a unit cell where none of the internal angles is 90 degrees.