Crystal Structure Explorer

The three cubic Bravais lattices, simple (SC), body-centred (BCC) and face-centred (FCC), with their conventional cells, a chosen Miller plane $(hkl)$ at spacing $d=a/\sqrt{h^2+k^2+l^2}$, and the powder X-ray pattern. The atom basis sets the geometric structure factor, so some reflections vanish: BCC needs $h+k+l$ even, FCC needs $h,k,l$ all even or all odd, which is why their powder lines follow the sequences 2,4,6,8 and 3,4,8,11. The reciprocal lattice (defined by $\vec b_i\cdot\vec a_j=2\pi\delta_{ij}$) is FCC for a BCC crystal and BCC for an FCC one, so the first Brillouin zones are the rhombic dodecahedron (12 faces) and the truncated octahedron (14 faces).