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).

Figure 1. Cubic crystal cell with a Miller plane, the reciprocal lattice / Brillouin-zone face count, and the powder XRD pattern. Method: closed-form lattice and reciprocal vectors; structure-factor selection rules; Bragg powder lines.

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Miller (hkl)

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WHAT TO TRY

Switch the lattice between SC, BCC and FCC: the ball-and-stick cell rebuilds with the right basis atoms, and the powder XRD pattern below changes its allowed peaks. FCC drops every mixed-parity reflection, BCC every h+k+l odd one.
Change the Miller indices (hkl): the gold plane reorients through the crystal and the matching XRD peak lights up gold. The d-spacing readout follows d = a / sqrt(h squared + k squared + l squared).
Switch the view to the reciprocal lattice or the single-crystal pattern: the same structure appears as its reciprocal-space dual, with the Brillouin-zone face and the (hkl) reflection spot, the bridge between real space and diffraction.