The unit cells of the solutions to the Kelvin problem and the corresponding members (grey balls) of the Kelvin carbons, including space groups and the nomenclature for polyhedra. Each polyhedron consists of three numbers [Q-P-H] that represent the number of quadrilaterals (Q), pentagons (P), and hexagons (H).
Binding energy as a function of volume for (a) diamond, lonsdaleite, and three-dimensional four-connected carbon allotropes recently discussed in the literature, and (b) Kelvin carbons.
Calculated binding energy (E b in eV/atom), equilibrium density (ρ in g/cm3), average bond length (d in Å), bandgap (E g in eV), bulk modulus (B 0 in GPa), and intrinsic hardness (H in GPa) for graphite, diamond (calculated and experimental), lonsdaleite, and recent hypothetical carbon structures (bct-C4, W-carbon, M-carbon, T-carbon, etc.). The corresponding calculated values are listed for the 11 Kelvin carbons shown in Fig. 1 .
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