Wulff construction

The Wulff construction

At fixed volume, a crystal in equilibrium takes the shape with the lowest total surface energy, , so expensive faces are kept small or removed. Wulff's theorem gives that shape directly: it is the region inside every facet plane, with each plane at a distance from the centre proportional to its surface energy, .

• Low-energy facets sit close to the centre, so they are cut largest.
• High-energy facets sit further out and shrink, or vanish if other planes already enclose them.
• A liquid has the same γ in every direction, so its shape is a circle or sphere.

Cubic crystals give cubes, octahedra, rhombic dodecahedra and everything in between. Lower-symmetry crystals, such as orthorhombic ones (rectangular in 2D), give each axis its own energies, which is how needles, laths and plates form. The same construction with growth rates in place of γ gives the kinetic growth shape: there too the slowest-growing faces end up largest.