How many distinct Bravais lattices exist in three dimensions?

• A. Twelve
• B. Fourteen
• C. Sixteen
• D. Eighteen

Answer: (B)

The question tests how many distinct ways a 3D lattice can be arranged so that every lattice point has the same environment, accounting for different lattice systems and how the points can be centered within the unit cell. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, trigonal (rhombohedral), and cubic. For each system, the lattice can be centered in several ways, and those centered variants that cannot be transformed into each other by symmetry are counted as separate Bravais lattices.

Triclinic has only one centering option (primitive). Monoclinic has two (primitive and base-centered). Orthorhombic has four (primitive, base-centered, body-centered, and face-centered). Tetragonal has two (primitive and body-centered). Hexagonal has one (primitive). Trigonal (rhombohedral) is represented by a single rhombohedral lattice. Cubic has three (primitive, body-centered, and face-centered). Adding these up gives 1 + 2 + 4 + 2 + 1 + 1 + 3 = 14 distinct Bravais lattices.

That combination of seven systems with their allowable centering patterns is what yields fourteen total, not fewer or more.