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Basic Program Features
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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

    1. The Simple Cube :
      Crystal system: Cubic
      Point Group: Any! Facets: [100]
      Energy: 1.0
      Wulff shape : Cube

    2. Truncated Octahedron :
      Crystal systems: Cubic
      Point Group: m3m (for example)
      Facets: [100] and [111]
      Energies: 1.0 and 0.85
      Wulff shape : Cuboctohedron

    3. 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

    4. 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

    5. Buckyballs! :
      Crystal system: Icosahedral
      Point Group: 235
      Facets: [111], [1 1.62 0]
      Energies: 1.0, 1.02
      Wulff shape : Buckyball

    6. 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
    1. 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)

    2. 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
    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
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Last updated: Sep 29, 2002