Form Factor:Superball

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A superball is a general mathematical shape that can be used to describe rounded cubes. In fact, it is a general parametrization that can describe, via a parameter :

  • Empty space ()
  • Concave octahedra ()
  • Octahedra ()
  • Convex octahedra ()
  • Spheres ()
  • Rounded cubes ()
  • Cubes ()

Superball examples.png

The general equation is parametrized by the size, , and the curvature :

Obviously for , we recover the equation for a sphere. In the limit of large , we obtain a cube.


The normalized volume for a superball is:

Where and is the usual Euler gamma function.

Superball volume.png


The form factor for a superball is likely not analytic. However, it can be computed numerically.


Mathematical descriptions of superballs

Application to nanoscience

Use in scattering