Visualize the Invisible Architecture of Materials

A Bravais lattice is one of 14 distinct three-dimensional lattices that describe every possible periodic arrangement of points in a crystal. They group into 7 crystal systems — cubic, tetragonal, orthorhombic, hexagonal, trigonal (rhombohedral), monoclinic, and triclinic — and form the geometric foundation of crystallography.

Body-Centered Cubic (BCC) places one atom at each cube corner plus one at the body centre, giving 2 atoms per unit cell, a coordination number of 8, and 68% packing efficiency. Face-Centered Cubic (FCC) has atoms at each corner and on every face, giving 4 atoms per unit cell, a coordination number of 12, and 74% packing efficiency — the densest cubic packing possible.

APF is the fraction of a unit cell's volume occupied by atoms, modelled as hard spheres. Common values: SC = 52%, BCC = 68%, FCC = HCP = 74%. Higher APF means a denser, more efficiently packed structure. You can compute APF interactively in our APF calculator.

Miller indices are a set of three integers (h k l) that uniquely identify a plane or direction within a crystal lattice. They are the reciprocals of the plane's intercepts with the crystal axes, reduced to lowest integer terms. Miller indices are essential for X-ray diffraction analysis, slip-system identification, and surface science.

A Brillouin zone is the primitive cell of the reciprocal lattice — the set of points in k-space closest to a given reciprocal-lattice point. The first Brillouin zone is fundamental in solid-state physics for describing electron band structure, phonon dispersion, and the behaviour of waves inside a periodic crystal.

Bragg's Law (nλ = 2d sin θ) relates the wavelength λ of incident X-rays, the interplanar spacing d, and the diffraction angle θ that produces constructive interference. It is the cornerstone of X-ray diffraction (XRD), letting researchers determine crystal structures from diffraction patterns.

Yes. All 14 Bravais lattices, the materials database, and the calculators (Bragg's Law, Miller indices, APF, band gap, symmetry) are free in your browser — no install, no account required. Optional sign-in lets you save favourites and use Crystal AI.

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