View source for Form Factor:Superball

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 <math>p</math>:
* Empty space (<math>p=0.0</math>)
* Concave octahedra (<math>p<0.5</math>)
* [[Form Factor:Octahedron|Octahedra]] (<math>p=0.5</math>)
* Convex octahedra (<math>0.5<p<1</math>)
* [[Form Factor:Sphere|Spheres]] (<math>p=1</math>)
* Rounded cubes (<math>p>1</math>)
* [[Form Factor:Cube|Cubes]] (<math>p \to \infty</math>)


[[Image:Superball examples.png|700px]]

The general equation is parametrized by the size, <math>R</math>, and the curvature <math>p</math>:
:<math>
\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}
</math>
Obviously for <math>p=1</math>, we recover the equation for a sphere. In the limit of large <math>p</math>, we obtain a cube.

==Volume==
The normalized volume for a superball is:
:<math>
\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)
</math>
Where <math>\mathrm{B}\left( x,y \right) = \Gamma(x)\Gamma(y)/\Gamma(x+y)</math> and <math>\Gamma(x)</math> is the usual [http://en.wikipedia.org/wiki/Gamma_function Euler gamma function].

[[Image:Superball volume.png|600px]]

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

==References==
====Mathematical descriptions of superballs====
* N. D. Elkies, A. M. Odlyzko and J. A. Rush "[http://www.springerlink.com/content/l481484244n16157/ On the packing densities of superballs and other bodies]" Inventiones Mathematicae Volume 105, Number 1 (1991), 613-639, [http://dx.doi.org/10.1007/BF01232282 doi: 10.1007/BF01232282]
* Y. Jiao, F.H. Stillinger, S. Torquato "[http://pre.aps.org/abstract/PRE/v79/i4/e041309 Optimal packings of superballs]" ''Physical Review E'' '''2009''', 79, 041309, [http://dx.doi.org/10.1103/PhysRevE.79.041309 doi: 10.1103/PhysRevE.79.041309]

====Application to nanoscience====
* Yugang Zhang, Fang Lu, Daniel van der Lelie, Oleg Gang "[http://prl.aps.org/abstract/PRL/v107/i13/e135701 Continuous Phase Transformation in Nanocube Assemblies]" ''Physical Review Letters'' '''2011''', 107, 135701 [http://dx.doi.org/10.1103/PhysRevLett.107.135701 doi: 10.1103/PhysRevLett.107.135701]

====Use in scattering====
* [[Yager, K.G.]]; Zhang, Y.; Lu, F.; Gang, O. "[http://scripts.iucr.org/cgi-bin/paper?S160057671302832X Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems]" ''Journal of Applied Crystallography'' '''2014''', 47, 118–129. [http://dx.doi.org/0.1107/S160057671302832X doi: 10.1107/S160057671302832X]
** See also [[Paper:DNA-nanoparticle superlattices formed from anisotropic building blocks|summary of paper]].