Editing Hard superball model
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The '''hard superball model''' is defined by the inequality | The '''hard superball model''' is defined by the inequality | ||
:<math>|x|^{2q} + |y|^{2q} +|z|^{2q} \le | :<math>\left|\frac{x}{a}\right|^{2q} + \left|\frac{y}{a}\right|^{2q} +\left|\frac{z}{a}\right|^{2q} \le 1</math> | ||
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 [[Hard cube model |cube]] (''q'' = ∞) via the [[Hard sphere model |sphere]] (''q'' = 1) as shown in the | 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 [[Hard cube model |cube]] (''q'' = ∞) via the [[Hard sphere model |sphere]] (''q'' = 1) as shown in the left figure. | ||
== Particle Volume == | == Particle Volume == | ||
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:<math> | :<math> | ||
\begin{ | \begin{eqnarray} | ||
v(q,a) & = & 8 a^3 \int_{0}^1 \int_{0}^{(1-x^{2q})^{1/2q}} (1-x^{2q}-y^{2q})^{1/2q} \mathrm{d}\, y \, \mathrm{d}\, x = \frac{ | v(q,a) & = & 8 a^3 \int_{0}^1 \int_{0}^{(1-x^{2q})^{1/2q}} (1-x^{2q}-y^{2q})^{1/2q} \mathrm{d}\, y \, \mathrm{d}\, x \nonumber\\ | ||
\end{ | & = & \frac{8a^3\left[ \Gamma\left(1+1/2q\right) \right]^3}{\Gamma\left(1+ 3/2q\right)}, | ||
\end{eqnarray} | |||
</math> | </math> | ||
where <math>\Gamma</math> is the [[Gamma function]]. | where <math>\Gamma</math> is the [[Gamma function]]. | ||
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==Phase diagram== | ==Phase diagram== | ||
The full [[phase diagrams |phase diagram]] of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref <ref name="superballs"></ref>. | The full [[phase diagrams |phase diagram]] of hard superballs whose shape interpolates from cubes to octahedra was reported in Ref <ref name="superballs"> </ref>. | ||
==References== | ==References== |