Crystalline Order on a Sphere and the Generalized Thomson Problem

Abstract

We attack generalized Thomson problems with a continuum formalism which exploits a universal long range interaction between defects depending on the Young modulus of the underlying lattice. Our predictions for the ground state energy agree with simulations of long range power law interactions of the form 1/r γ (0 < γ < 2) to four significant digits. The regime of grain boundaries is studied in the context of tilted crystalline order and the generality of our approach is illustrated with new results for square tilings on the sphere.PACS numbers: PACS numbers: 68.35. Gy, 62.60.Dc, 61.72.Lk. 61.30.Jf The Thomson problem of constructing the ground state of (classical) electrons interacting with a repulsive Coulomb potential on a 2-sphere [1] is almost one hundred years old [2] and has many important physical realizations. These include multi-electron bubbles [3], which may be studied by capillary wave excitations, or the surface of liquid metal drops confined in Paul traps [4]. Although the original Thomson problem refers to the ground state of spherical shells of electrons, one can also ask for crystalline ground states of particles interacting with other potentials. Such a generalized Thomson problem arises, for example, in determining the arrangements of the protein subunits which comprise the shells of spherical viruses [5,6]. Here, the "particles" are clusters of protein subunits arranged on a shell. Other realizations include regular arrangements of colloidal particles in colloidosomes [7] proposed for encapsulation of active ingredients such as drugs, nutrients or living cells [8] and fullerene patterns of carbon atoms on spheres [9] and other geometries [10]. An example with long range (logarithmic) interactions is provided by the Abrikosov lattice of vortices which would form at low temperatures in a superconducting metal shell with a large monopole at the center [11]. In practice, the "monopole" could be approximated by the tip of a long thin solenoid.Extensive numerical studies of the Thomson problem show that the ground state for a small number of particles, typically M ≤ 150, consists of twelve positive disclinations (the minimum number compatible with Euler's theorem) located at the vertices of an icosahedron [12,13]. Recent results have shown that for systems as small as 500 particles, however, configurations with additional topological defects [14,15] have lower energies than icosahedral ones.These remarkable results for the Thomson problem raise a number of important questions, such as the mechanism behind the proliferation of defects, the nature of these unusual low-energy states, the universality with respect to the underlying particle potential and the generalization to more complex situations.A formalism suitable to address all these questions has been proposed recently [16]. Disclinations are considered the fundamental degrees of freedom, interacting according to the energy [16]where the integration is over a fixed surface with area element dσ(x) and metric g ij , K is the Gaussian curvature...

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