# Hard superball model

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The **hard superball model** is defined by the inequality

where *x*, *y* and *z* are scaled Cartesian coordinates with *q* the deformation parameter and radius *a*. The shape of the superball interpolates smoothly between two Platonic solids, namely the octahedron (*q* = 0.5) and the cube (*q* = ∞) via the sphere (*q* = 1) as shown in the right figure.

## Particle Volume[edit]

The volume of a superball with the shape parameter *q* and radius *a* is given by

where is the Gamma function.

## Overlap algorithm[edit]

The most widely used overlap algorithm is on the basis of Perram and Wertheim method ^{[1]} ^{[2]}.

## Phase diagram[edit]

The full phase diagram of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref ^{[2]}.

## References[edit]

- ↑ John W. Perram and M. S. Wertheim "Statistical mechanics of hard ellipsoids. I. Overlap algorithm and the contact function", Journal of Computational Physics
**58**pp. 409-416 (1985) - ↑
^{2.0}^{2.1}R. Ni, A.P. Gantapara, J. de Graaf, R. van Roij, and M. Dijkstra "Phase diagram of colloidal hard superballs: from cubes via spheres to octahedra", Soft Matter**8**pp. 8826-8834 (2012)