Examples Wulffman is a versatile tool capable of illustrating an infinite variety of crystal shapes. A few examples are given below that demonstrate some of the basic program features listed above. Simple Wulff Shape Construction The Simple Cube : Crystal system: Cubic Point Group: Any! Facets: [100] Energy: 1.0 Wulff shape : Cube Truncated Octahedron : Crystal systems: Cubic Point Group: m3m (for example) Facets: [100] and [111] Energies: 1.0 and 0.85 Wulff shape : Cuboctohedron A hexagonal Pencil : Crystal system: Hexagonal Point Group: 6/mmm Facets: [431], [100], [210] Energies: 1.0, 0.86, 0.91 Wulff shape : Modified Dihexagonal Dipyramid A trigonal Lozenge : Crystal system: Trigonal Point Group: 3_m Facets: [14n], n=6-12 Energies: 0.65, 0.59, 0.55, 0.52, 0.50, 0.485, 0.48 Wulff shape : Modified Hexagonal Scalenohedron Buckyballs! : Crystal system: Icosahedral Point Group: 235 Facets: [111], [1 1.62 0] Energies: 1.0, 1.02 Wulff shape : Buckyball Custom Flying Saucer : Crystal system: Custom Point Group: 37-fold roto-inversion around [001] Facets: [148], [100], [217], [124] Energies: 0.85, 1.63, 0.87, 0.85 Wulff shape : Flying saucer (?) Dynamic Wulff Shapes : Surface energy anisotropy Octahedron --> Truncated Octahedron --> Dodecahedron: Crystal system: Cubic Point group: m3_m Transformation: The beginning structure is an octahedron generated by [111] facets. [100] facets are added, and their surface energy is lowered until the truncated octahedron (cuboctahedron) results. [110] facets with low energy are included, and as their energy is decreased relative to [100] and [111], the dodecahedron results. Graphics: Animated GIF (170k) Stars and Icosahedra Forever: Crystal system: Icosahedral Point group: 235 Transformation: A general icosahedral form with 60 [132] facets is generated. [111] facets are included and their energy is decreased until the regular icosahedra results. Graphics: Animated GIF (270k) Unique Crystal Planes: Crystal system: Cubic Point group: 432 Facets: Unique ("slice") plane: [112] Normal facets: [149], [216], [100] Description: Decreasing the energy of the unique plane [112] progressively slices off more and more of the Wulff shape. If the [112] facet had not been unique, 24 equivalent planes would have been generated. Graphics: Still GIF (11k) and Animated GIF (308k) Isotropic Surfaces Crystal system: Cubic Point group: m3_m Facets: [100] (Energy = 0.85) Boundary polytope: 500 facets, Skew = 0, Energy = 1.0 Description: In the absence of a bounding polytope, [100] facets under cubic symmetry generate a cube Wulff shape. By adding an isotropic boundary polytope with a slightly higher energy, the corners end edges of the cube are cut off and replaced by smooth surfaces. The Wulff shape is effectively the intersection of a sphere and a cube. Graphics: Still GIF (30k) and animated GIF (335k) Naturally-Occurring Materials The following are examples of Wulff shapes found to occur in nature (from Elementary Crystallography , M. Buerger, MIT Press, 1956): Sulfur: An example of development in class mmm Eglestonite : Hg4OCl2, an example of form development in class 43_m. Beryl : Be3Al2Si6O18, Beryllium Aluminum Silicate, a semi-precious mineral, represents form development in class 6mm. Rutile : TiO2, Titanium Dioxide, an example of form development in class 4mm. Center for Theoretical and Computational Materials Science, NIST Questions or comments: wulffman@ctcms.nist.gov
 NIST is an agency of the U.S. Department of Commerce

Last updated: Sep 29, 2002