Defect motifs for spherical topologies

Putative global minima for the Thomson problem of $N$ charges on a sphere are located for selected sizes in the range $N\le 4352$. For small sizes most global minima simply exhibit 12 disclinations. However, around $N=400$ extended dislocations are common, where a heptagon lies between two pentagons in the Voronoi representation. At larger sizes around $N=1000$ twinned grain boundaries are favorable and then between about $N=1000$ and 2000 “rosette” defects appear, containing a central pentagon surrounded by five heptagons alternating with five additional pentagons in the Voronoi representation. Structures with 12 rosettes and $I$ point group symmetry are particularly low in energy for $N=1632$ and 1902. Above $N=2000$ the lowest minima located generally contain irregular grain boundaries, sometimes with a few rosette defects. Face dual carbon clusters corresponding to $I_h$ local minima at $N=432$ and 582 are characterized as local minima for ${\text{C}}_{860}$ and ${\text{C}}_{1160}$.

Figure 1

(Color online) Voronoi representations of the lowest minima located for $N=400,410,\dots ,570$ and $N=1500$, 2000, and 2500. The pentagons, hexagons, and heptagons are colored red (medium gray), green (light gray), and blue (dark gray), respectively.

Figure 2

(Color online) Voronoi representations of the lowest minima located for selected sizes at which local minima with $I$ or $I_h$ symmetry exist for structures with 12 rosettes. The pentagons, hexagons, and heptagons are colored red (medium gray), green (light gray), and blue (dark gray), respectively.

Figure 3

(Color online) Plot of the number of pentagons $(+)$ and heptagons $(\times )$ in the Voronoi representations of the lowest known minima for the Thomson problem. Results for $N\le 400$ are taken from the structures reported in Ref. 39. The results for $N>400$ correspond to global optimization results for the sizes in Table where stable minima exist with $I$ or $I_h$ symmetry for the ${\left(0,k\right)}^{-}$ series based on 12 rosette defects, along with the results from $N=400$ to $N=570$ in Table .

Figure 4

(Color online) Geometry-optimized ${\text{C}}_{860}$ (top) and ${\text{C}}_{1160}$ (bottom) clusters. Views are shown down the approximate $C_2$ (left) and $C_5$ (right) axes with pentagonal rings highlighted in red (light gray).