Reciprocal lattice: Let a1, a2, and a3 be the primitive vectors of a Bravais lattice, and b1, b2, and b3 be the primitive vectors of the corresponding reciprocal lattice.
Using the construction:
b1 = 2π (a2 × a3) / (a1 · (a2 × a3))
b2 = 2π (a3 × a1) / (a1 · (a2 × a3))
b3 = 2π (a1 × a2) / (a1 · (a2 × a3))
Show that the reciprocal lattice of the face-centered cubic (fcc) lattice is the body-centered cubic (bcc) lattice with lattice spacing 4π/a and vice versa. Find the reciprocal lattice vectors of smallest magnitude for simple cubic (sc), fcc, and bcc lattices, and write down their Miller indices. Show that these correspond to planes in the direct lattice whose distance is d = 2π / |b|.