Crystalline particle packings on a sphere with long-range power-law potentials

The original Thomson problem of "spherical crystallography" seeks the ground state of electron shells interacting via the Coulomb potential; however one can also study crystalline ground states of particles interacting with other potentials. We focus here on long-range power-law interactions of the form 1 rγ (0<γ<2), with the classic Thomson problem given by γ=1. At large Ra, where R is the sphere radius and a is the particle spacing, the problem can be reformulated as a continuum elastic model that depends on the Young's modulus of particles packed in the plane and the universal (independent of the pair potential) geometrical interactions between disclination defects. The energy of the continuum model can be expressed as an expansion in powers of the total number of particles, M∼ (Ra)2, with coefficients explicitly related to both geometric and potential-dependent terms. For icosahedral configurations of 12 five-fold disclinations, the first nontrivial coefficient of the expansion agrees with explicit numerical evaluation for discrete particle arrangements to four significant digits; the discrepancy in the fifth digit arises from a contribution to the energy that is sensitive to the particular icosadeltahedral configuration and that is neglected in the continuum calculation. In the limit of a very large number of particles, an instability toward grain boundaries can be understood in terms of a "Debye-Huckel" solution, where dislocations have continuous Burgers' vector "charges." Discrete dislocations in grain boundaries for intermediate particle numbers are discussed as well.