Questions: Crystal Structure and Bravais Lattices

Question 1

A Bravais lattice requires that every lattice point have an identical environment when viewed from that point. Which of the following is NOT a Bravais lattice?

• A. Simple cubic
• B. Body-centered cubic (BCC)
• C. Honeycomb lattice
• D. Face-centered cubic (FCC)

Answer: (C)

The honeycomb lattice (like graphene) is not a Bravais lattice because it contains two inequivalent sites per unit cell. It is instead described as a hexagonal Bravais lattice with a two-atom basis. Every Bravais lattice must look identical from every point, which fails for the honeycomb because alternating vertices have different neighbor configurations.

Question 2

What distinguishes a primitive unit cell from a conventional unit cell in crystallography?

• A. A primitive cell always has cubic symmetry; a conventional cell does not
• B. A primitive cell contains exactly one lattice point and is the smallest repeating unit, while a conventional cell may contain multiple lattice points but is chosen to display the full symmetry of the lattice
• C. A primitive cell and conventional cell always have the same volume
• D. A conventional cell is always smaller than a primitive cell

Answer: (B)

The primitive unit cell contains exactly one lattice point (volume = |a_1 · (a_2 × a_3)|) and tiles space without gaps. However, primitive cells can have awkward shapes that obscure the lattice symmetry. The conventional cell is chosen to make symmetry obvious — for example, the conventional BCC cell is a cube containing 2 lattice points, while the primitive BCC cell is a rhombohedron. The conventional cell is always an integer multiple of the primitive cell volume.

Question 3

The 14 Bravais lattices in 3D arise because some crystal systems allow centering (body-centered, face-centered, base-centered) while others do not, since centering would produce a lattice equivalent to a different type.

• T. True
• F. False

Answer: True

Starting from the 7 crystal systems (defined by axial lengths and angles), one can attempt to add centering points. But many centerings are redundant — for instance, a base-centered cubic lattice is equivalent to a tetragonal lattice after redefining axes. After eliminating all such redundancies, exactly 14 distinct lattices remain. This classification was established by Auguste Bravais in 1850.

Question 4

Explain why the diamond structure requires a basis of two atoms on an FCC lattice, rather than being a Bravais lattice itself.

Model answer: In the diamond structure, each atom is tetrahedrally bonded to four neighbors, but the two atoms in each pair have different orientations of their bonding environment. Specifically, the structure consists of two interpenetrating FCC sublattices offset by (a/4, a/4, a/4). An atom on one sublattice sees its neighbors arranged differently than an atom on the other sublattice would if you simply translated the lattice. Since a Bravais lattice demands that every point has an identical environment, diamond fails this test and must be described as FCC + a two-atom basis.

This is a common source of confusion. Diamond, silicon, and germanium all have this structure. The key insight is that 'identical environment' in the Bravais lattice definition means identical under pure translation — no rotation or reflection allowed. The two sublattice sites in diamond are related by a non-primitive translation, not by any lattice vector.