Hard superball model

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The shape of superballs interpolates between octahedra (q = 0.5) and cubes (q = ∞) via spheres (q = 1).
Phase diagram for hard superballs in the (packing fraction) versus 1/q (bottom axis) and q (top axis) representation where q is the deformation parameter [2].

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].