Form Factor:Superball

From GISAXS
Revision as of 11:38, 20 January 2015 by KevinYager (talk | contribs) (References)
Jump to: navigation, search

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.

Volume

The normalized volume for a superball is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{ V_{\mathrm{sb}} }{R^3} = \frac{2}{2p} \mathrm{B}\left( \frac{1}{p} , \frac{2p+1}{2p} \right) \mathrm{B}\left( \frac{1}{2p} , \frac{p+1}{p} \right) }

Where and is the usual Euler gamma function.

Superball volume.png

Equations

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

References

Mathematical descriptions of superballs

Application to nanoscience

Use in scattering