Abstract
In a previous report, the approximate crystalline structure and electronic structure of a novel, hypothetical hexagonal carbon allotrope has been disclosed. Employing the approximate extended Hückel method, this C structure was determined to be a semi-conducting structure. In contrast, a state-of-the-art density functional theory (DFT) optimization reveals the hexagonal structure to be metallic in band profile. It is built upon a bicyclo[2.2.2]-2,5,7-octatriene (barrelene) generating fragment molecule, and is a Catalan network, with the Wells point symbol (66)2(63)3 and the corresponding Schläfli symbol (6, 3.4). As the network is entirely composed of hexagons and, in addition, possesses hexagonal symmetry, lying in space group P6/mmm (space group #191), it has been given the name hexagonite. The present report describes a density functional theory (DFT) optimization of the lattice parameters of the parent hexagonite structure, with the result giving the optimized lattice parameters of a = 0.477 nm and c = 0.412 nm. A calculation is then reported of a simple diffraction pattern of hexagonite from these optimized lattice parameters, with Bragg spacings enumerated for the lattice out to fourth order. Results of a synchrotron diffraction study of carbon nanotubes which underwent cold compression in a diamond anvil cell (DAC) to 100 GPa, in which the carbon nanotubes have evidently collapsed into a hitherto unknown hexagonal C polymorph, are then compared to the calculated diffraction pattern for the DFT optimized hexagonite structure. It is seen that a close fit is obtained to the experimental data, with a standard deviation over the five matched reflections being given by σx = 0.003107 nm/reflection.
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Notes
- 1.
The C structure described in this communication, and elsewhere, with the name hexagonite is not to be confused with the inorganic mineral structure of the same name. The authors felt it appropriate to name the C structure, described herein, as hexagonite because of the special circumstance of its hexagonal symmetry space group (P6/mmm, #191), combined with its further 6-ness, as distinguished by its topological polygonality, given by n = 6, in which all smallest circuits in the network are hexagons.
- 2.
CASTEP (Cambridge Serial Total Energy Package) is a plane wave pseudopotential code, based upon density functional theory (DFT), that was used to optimize the hexagonite structure in the present report. Therefore, for the present implementation of CASTEP, used to optimize the structural parameters of hexagonite, the local density approximation (LDA) was used, ultrasoft pseudopotentials were employed, the basis set had an energy cutoff of 400 eV and k-point sampling was done with a 10×10×4 mesh. The ultrasoft pseudopotentials used in the calculation are due to Vanderbilt (D. Vanderbilt, “Soft Self-Consistent Pseudopotentials in a Generalized Eigenvalue Formalism,” Phys. Rev. B, 1990, 41 (Rapid Communications), 7892–7895.) The Brillouin zone was sampled at a density of 0.004 nm−1.
- 3.
The CASTEP-DFT method calculates the band structure of hexagonite to be metallic, in contrast the approximate EHMO method calculates the hexagonite structure to have a semi-conducting band profile. It is believed that the EHMO calculations of semi-conducting hexagonite are closer to a true reflection of the electronic structure of the lattice than that provided by the DFT results, based upon the fact that hexagonite can be viewed as a layering of delocalized π bonding (sp2) sandwiched between insulating layers of C σ bonding (sp3), and thus it cannot realistically be represented as a 3D metallic structure.
- 4.
The volume of the hexagonite unit cell was calculated with the formula V = ((3)1/2/2)(a 2 c), and the corresponding density of hexagonite, with 10 C atoms in the unit cell, was found to be 2.449 g/cm3. The density of the hexagonal C polymorph reported by Wang et al. in 2004 is 32% greater than this calculated value, at 3.6 g/cm3.
- 5.
On a stability scale at which diamond is at the 0 of energy, the glitter allotrope of C, described in [22], has an energy of formation of 0.5116 eV/C atom above that of diamond, by the CASTEP optimization, under the local density approximation (LDA). In contrast, the hexagonite allotrope of C, described herein, has an energy of formation of 0.5343 eV/C atom above that of diamond by the same computational method.
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Acknowledgements
MJB thanks his wife Hsi-cheng Shen for much love and patience in his work on C allotropy and the subtle structural issues of C. The authors wish to thank Norman Goldberg, PhD for producing the structural drawings of hexagonite while a post-doctoral associate in Professor Roald Hoffmann’s theoretical chemistry group at Cornell University. The authors wish to thank Chris J. Pickard, PhD of the Theoretical Condensed Matter (TCM) Group at Cambridge University, for his great help in carrying out the DFT-CASTEP optimization calculations of the hexagonite structure. The authors wish to thank Roald Hoffmann for his suggestions in writing this manuscript. Finally, the authors wish to thank D.M.E. (Marian) Szebenyi, PhD at Cornell High Energy Synchrotron Source (CHESS) for helpful discussions of the symmetry aspects of hexagonite.
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Bucknum, M.J., Castro, E.A. (2011). High Pressure Synthesis of the Carbon Allotrope Hexagonite with Carbon Nanotubes in a Diamond Anvil Cell. In: Putz, M. (eds) Carbon Bonding and Structures. Carbon Materials: Chemistry and Physics, vol 5. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-1733-6_5
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