 Solution

For conventional cubic cell side $a$, the face-centered cubic lattice has corner and face-centre sites, giving four lattice points per cell; the body-centered cubic lattice has corners and a body centre, giving two. Their primitive unit cells each contain one lattice point. With reciprocal vectors written $\mathbf{G}=(2\pi /a)(h,k,l)$, face-centered cubic lattice allows $h,k,l$ all even or all odd, while body-centred cubic allows $h+k+l$ even. HenceHere $a^{\ast }$ is the side of the reciprocal lattice's conventional cubic cell. The parity conditions describe the reciprocal point sets; reciprocal space contains wavevectors, not a second arrangement of physical atoms.