Electrostatic energy of a one-component quasicrystal

The Madelung constant associated with the electrostatic energy of perfect one-component icosahedral quasicrystals is evaluated using various simple connected acceptance volumes. Analytic forms for the structure factors and pair probabilities are used for both spherical and polyhedral acceptance volumes. The maximum value for the quasicrystal Madelung constant with a spherical acceptance volume is 1.623, and for the three-dimensional Penrose tiling it is 1.655. The acceptance volumes that give the highest hard-sphere packing fractions are subsets of this Penrose tiling at various scales. The best convex ones are either an icosahedron or the intersection of a dodecahedron with a triacontahedron, and these are found to have Madelung constants of 1.602 and 1.712, respectively. The two corresponding nonconvex volumes that give the best possible sphere packings have Madelung constants of 1.742 and 1.717, respectively, and so would be the most likely candidates for the structure of a one-component quasicrystalline metal. Convergence of the Madelung sums is much slower in these quasicrystal systems than in regular crystals, and the results are checked by two other approximations, showing that the numbers quoted above have converged to within about ${10}^{\mathrm{-}3}$.