Question

Consider the conventional bcc unit cell, choose an origin at (000) and determine the shortest translational vectors required to define the lattice. Show that the associated reciprocal lattice has fcc structure.

Answer 1

Primitive translation vectors of the bcc lattice (in units of lattice parameter a) are a₁ = ½ ½ -½; a₂ = - ½ ½ ½; a₃ = ½ -½ ½. The primitive cell is the rhombohedron. The packing ratio is 0.68, defined as the maximum volume which can be filled by touching hard spheres in atomic positions. Each atom has 8 nearest neighbors.

The conventional unit cell is a cube based on vectors a₁ = 001; a₂ = 010; a₃ = 001. It is twice big compared to the primitive unit cell and has two atoms in it with coordinates r₁ = 000 and r₂ = ½ ½ ½.

Reciprocal lattice to bcc lattice:

By cyclic symmetry, one can deduce:

Clearly, , , generates a FCC Bravais lattice. Therefore, the reciprocal lattice to bcc is the fcc lattice. It is straightforward to show that the reciprocal lattice of fcc lattice is the bcc lattice.