  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/r^gamma (0 < \gamma < 2), with the classic Thomson problem given by gamma=1.
At large R/a, 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 sim (R/a)^2, with coefficients explicitly
related to both geometric and potential-dependent terms. For icosahedral
configurations of twelve 5-fold disclinations, the first non-trivial
coefficient of the expansion agrees with explicit numerical evaluation for
discrete particle arrangements to 4 significant digits; the discrepancy in the
5th 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.
