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 p:

  • Empty space (p=0.0)
  • Concave octahedra (p<0.5)
  • Octahedra (p=0.5)
  • Convex octahedra (0.5<p<1)
  • Spheres (p=1)
  • Rounded cubes (p>1)
  • Cubes (p \to \infty)


Superball examples.png

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


\begin{alignat}{2}
\left | \frac{x}{R} \right | ^{2p} + \left | \frac{y}{R} \right | ^{2p} + \left | \frac{z}{R} \right | ^{2p} & \le 1 \\
| x | ^{2p} + | y | ^{2p} + | z | ^{2p} & \le |R|^{2p} \\
\end{alignat}

Obviously for p=1, we recover the equation for a sphere. In the limit of large p, we obtain a cube.

Volume

The normalized volume for a superball is:


\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 \mathrm{B}\left( x,y \right) = \Gamma(x)\Gamma(y)/\Gamma(x+y) and \Gamma(x) 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