Phys. Status Solidi B, 1–5 (2012) / DOI 10.1002/pssb.201248185 p s sb

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basic solid state physics

‘‘C8 phase’’: Supercubane,tetrahedral, BC-8 or carbon sodalite?

Alex Pokropivny1,2, and Sebastian Volz*,1

1 Laboratoire d’Energetique Moleculaire et Macroscopique, Combustion, UPR CNRS 288, Ecole Centrale Paris,

Grande Voie des Vignes, 92295, Chatenay Malabry, France2 Frantsevich Institute for Problems of Materials Science NASU, Krzhizhanovsky str., 3, 03680, Kiev, Ukraine

Received 1 February 2012, revised 29 May 2012, accepted 6 June 2012

Published online 00 Month 2012

Keywords Ab initio DFT calculations, Diamond, Sodalite, Unconventional carbon phases, X-ray and electron diffraction

* Corresponding author: e-mail sebastian.volz@ecp.fr, Phone: þ33 14113 1070

With the use of ab initio DFT and electron diffraction

calculations, the crystal structures of a carbon ‘‘C8 phase’’ are

found. Total energies and electron diffraction patterns are

calculated for all possible known to date candidates, namely

supercubane, tetrahedral, BC-8, and sodalite (SOD) structures

as nearest neighbours in unit cell size. The results of calculation

agree well with only one phase. The structure can be attributed

to sp3-hybridized carbon SOD zeolite with 12 atoms in a unit

cell. Carbon sodalite belongs to space group No. 229, with a

unit cell of a¼ 4.34 A, and a calculated crystal density of

2.927 g/cm3. The existence of a tetrahedral phase (a¼ 3.86 A)

is discussed. A simple formula of relative structural stability

is proposed for all sp3-hybridized phases with unit cells of

3.5–5.0 A.

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1 Introduction Carbon is an unique life element andbecause of its different hybridization types it is a well-knownbuilder. A lot of experimental and theoretical efforts weremade up to now for finding new unconventional phases ofcarbon and boron nitride, as structural analogue [1–3].Synthesized so-called ‘‘C8 carbon’’ is one of such unconven-tional allotropes, wich crystal structure has been discussedfor more than 20 years [4–7]. Crystals surely belong to thecubic symmetry with a lattice parameter of 4.28–4.29 A forpolycrystals and 4.34 A for monocrystals. The difficulty instructure determination of such carbons is attributed to therelatively small yields by mass and small sizes (nanometers)in comparison to other phases, like diamond, graphite orfullerites and nanotube crystals.

In this article we will resolve the structure of this uniquephase in the family of carbons on the basis of the analysis ofavailable experimental data, and by making some compara-tive simulations. To this aim, several associated structures

and electron diffraction patterns are simulated with the samemethod to find their relative stability and structure factors.

Matyushenko at al. [4] have observed the new phase forthe first time in 1979 in the products of condensation ofcarbon plasma flow in vacuum. The films of this new phasewere homogeneously deposited and saved on cool surfaces.The six diffusion rings, namely, (011), (002), (112), (022),(013), and (222) on micro electron-diffraction patterns ofpolycrystalline structure were indexed in the cubic systemwith a bcc lattice period of 4.28 A. Additionally, single-crystal morphologically formed structures were observedin dispersed carbon films with sizes ranging from 100 A to3000 A by electron microscopy. The interplanar spacing wasalso indexed with a bcc structure of 4.34 A lattice constant, asdetermined by pairs of Kikuchi lines. The faceted crystalswere supposed to be identical by the crystal structure to thepolycrystalline formations. The structure was proposed asthe Im3 space group with supercubane structural type as

� 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

2 A. Pokropivny and S. Volz: ‘‘C8 phase’’: Supercubane, tetrahedral, BC-8 or carbon sodalite?p

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Figure 1 Three of eight different spot electron diffraction patternscorresponding to the zones h110i, h210i, and h311i for the samecarbon crystal [5].

Figure 2 Carbon sodalite clusters as calculated with rhf/6-31glevel of theory for C144H98 (a), and C168H96 (b). Large circles denotecarbon atoms, while small ones denote hydrogen atoms.

being built by eight-atom clusters, interconnected by singlebonds, with a calculated density of 4.1 g/cm3.

Later, in 1988, Kostadinov and Dobrev [5] probablyobserved the same carbon phase as small crystallites withcubic habitus in carbon layers formed under depositiontechnique when the growing layer on a graphite substrate wassubjected to intense bombardment by low energy argon ionsfrom a gas discharge plasma. Electron diffraction investi-gation of the crystallites indicated a lattice parameter of 4.29A, practically the same as for the polycrystalline species of‘‘C8 phase’’. Several different spot electron diffractionpatterns were obtained, three of those were available in theirpaper [5]. Authors claimed the new phase as having Pn3mspace group symmetry rather than Im3. The tetragonalstructure for atomic positions was proposed, as being built byfour-atom clusters, interconnected by single C–C bonds.

Nevertheless, both analysis of the structures were maderoughly without taking into account the carbon bondingnature. For example, first authors [4] received a theoreticaldensity of 15% greater than diamond without any referenceto experimental measurements of the density and reliabilityof the bonds to theoretical approaches. Second authors [5], intheir turn, just proposed a structure without any references onthe correspondence of the arrangement of carbon withexperimental ones, and the analysis of the allowed reflectionsremained uncompleted. Later the BC-8 structure wasproposed as a candidate for the ‘‘C8 phase’’ [6, 7] withoutany reference on the correspondence between theoreticaldiffraction patterns and experimental ones. Thus additionalanalysis should be provided for that purpose.

Recently, in 2008 Liu et al. [8] synthesized micro- andnanocubes of C8-like phase by laser ablation in liquid. Thesesingle crystals were attributed to the Matyushenko’sstructure but with a 5.46 A lattice cell. In this case theinterbond distances should be 1.722 A, which is out of rangeof diamond values. It is very strange, if not false, thatseparate cubes can form van-der-Waals-like interaction, asin fact reported by Liu et al. [8] in their calculations.

What is the crystal structure of C8? We analyze herethese sets of possible known to date structures with anappropriate unit cell length and propose a new structure likesodalite as a new promising candidate for that phase. All ofthem are well known geometrically, and have beencalculated for carbon and other compositions.

Crystal structures are calculated on the basis of Wyck-off’s positions of the appropriate space groups by S. Weber’sJSV code. Four structures, namely, supercubane [4],tetrahedral [5], BC-8 [6, 7], and sodalite [9, 10] are analyzed.Geometrically, carbon sodalite can be easily reconstructedfrom other carbon LTA zeolite [11–13] by replacinginterconnecting C8 cubes by C4 squares.

It should be noted that we have simulated the carbonsodalite clusters in 2008 in accordance to the DFG projectDE 412/35-1, as seen in Figure 2. The nanoclusters of newpossible zeolite-like phases, nanotube contacts, nanocrystalswith different orientations of building units were calculatedwith the quantum-chemistry method, using GAMESS code

� 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

at the rhf/6-31g level of theory. Clusters were constructed bymultiplating single C24 cages in different ways as a part of thezeolite-like frameworks. For sodalite cages the most stablestructures were found to be the structures with diamond-likebonds. The stability of the new phases are confirmed. Thebond lengths equal to 1.557 A (C144H98) and 1.553 A(C168H96) in the central ideal truncated (4.6.6)-C24 octa-hedrons.

The quantum-chemistry calculations helped us in theearly stage of crystal structure determination and could beused further in the other programs, including SIESTA orother periodic-boundary based programs. As we did notmanage to completely decipher crystal structures and failedin supercritical fluid synthesis of the crystals, suchcalculations were premature and were not published untilnow.

2 Method Crystal calculations are based on the abinitio density-functional theory with the Perdew, Burke, andErzernhof generalized gradient approximation for theexchange-correlation functional. Norm-conserved Troul-lier-Martins pseudopotential for carbon with the samefunctional are used as implemented in SIESTA code [14].Brillouin-zone integrations are performed with Monkhorst-Pack algorithm for a mesh of 8� 8� 8 k-point grid.Conjugate gradient minimization scheme with varying bothcell dimension and atomic positions is utilized duringmolecular dynamics relaxations of the structures. Thegeometries are optimized when the forces on the atoms

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Phys. Status Solidi B (2012) 3

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become smaller than 0.04 eV/A and the stress tensorbecomes less than 0.02 GPa. Calculations of the structurefactors are performed with GDIS computer code [15] on thebasis of the optimized unit cells.

3 Results Electron diffraction patterns (U¼ 64 V) forall optimized structures are shown in Figure 3, on the right ofthe appropriate cells. Relative intensities are fitted to theintensity of the main peak, which is the same for all sets ofcalculations.

The results of calculation are summarized in Table 1.Analysis of the experimental and theoretical data can be

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Figure 3 Optimized unit cells for carbon sodalite (a), BC-8 (b),subercubane (c), and tetrahedral (d) phases with correspondingcalculated electron diffraction patterns (in 2Q degrees scale).Larger fonts indicate experimentally observed patterns, smallerfonts denote reflections which are out of range for experimentaldata, and crosses denote experimental reflections which are notobserved [4].

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started to find the structure. Diamond is the most stablestructure, with the relative energy gap per atom ofDE¼ 0.406 eV in respect to carbon sodalite. Nevertheless,carbon sodalite is the more stable one between all of theseunconventional sp3-carbons with relative energy gaps of0.247 eV, 0.301 eV, and 1.072 eV, respectively. Thereby thisstructure can be experimentally synthesized.

Speaking in the terms of energy, the sodalite is preferableamong all other candidates, with the smallest deviation ofbond length from diamond value (less than 0.1%), seeTable 2.

With that point of view, supercubane becomes the mostfavorable structure after sodalite. As for angles, BC-8structure has the smaller angle deviation, but, simul-taneously, has also a total deviation of the bonds by morethan 10%. The tetrahedral and supercubane structures are theworst both by angle and bond deviations.

As a criterium of total angle and bond deformations ofthe ideal diamond tetrahedron, the arithmetical mean of bothangles and bonds deviations can be calculated, namely:16.1% and 2.75% for supercubane, 38.7% and 2.84% fortetrahedral carbon, 10.1% and 5.40% for BC-8 carbon,

Table 1 Space group symmetry (s.g.), number of atoms per unitcell (Z), lattice constant (l, A) and relative deviation to exper-imental values (ldev, %), calculated density (r, g/cm3), total energyper atom (DEtot, eV/atom), band gap energy (Eg, eV) for fivedifferent carbon allotrope phases.

Phase s.g. Z l ldev� i �Etot

�� Eg

Diamond 227 8 3.584 þ0.4 3.413 0 3.36Supercubane 204 16 4.889 þ12.6 2.730 0.653 2.36Tetrahedral 224 8 3.840 �11.5 2.818 1.478 0.25BC-8 206 16 4.491 þ3.5 3.522 0.707 1.71Sodalite 229 12 4.392 þ1.2 2.826 0.406 2.13

�in relation to experimental values a¼ 3.57 A (r¼ 3.515 g/cm3,

Eg¼ 5.45 eV) for diamond and a¼ 4.34 A for other phases. ��in relation to

the total energy of c-diamond (Etot¼�156.195 eV/atom).

Table 2 The calculated bonds (l, A) and angles (Q

, degrees) withcorresponding number of bonds (Nb) and angles (Na) per tetra-hedron, relative deviation of the bonds (ldev, %) and angles (

Qdev,

%) to the diamond values, and total deviation (D).

Phase D� Bonds Angles

l, A Nb ldevQ

Na

Qdev

Diamond 0 1.552 4 0 109.47 6 0Supercubane 0.726 1.486 1 �4.25 90 3 �17.8

1.587 3 þ2.26 125.26 3 þ14.4Tetrahedral 1.479 1.457 1 �6.12 60 3 �45.2

1.525 3 þ1.74 144.74 3 þ32.2BC-8 0.697 1.466 1 �5.54 97.92 3 �11.55

1.635 3 þ5.35 118.14 3 þ8.67Sodalite 0.414 1.553 4 þ0.06 90 2 �17.79

120 4 þ9.62

�calculated by formula (1).

� 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

4 A. Pokropivny and S. Volz: ‘‘C8 phase’’: Supercubane, tetrahedral, BC-8 or carbon sodalite?p

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12.3% and 0.06% for sodalite. From these sets, it is difficultto conclude on the relative stability. I.e., BC-8 structure isvery attractive if only angles are under consideration, while itis the worst, when bonds are taking into account. Obviously,both angle and bond deformations should be taken intoaccount for such purposes. Let us calculate the bond-anglerelative deformations numbers as the mean square of a half ofthe first and the second values. Thus, we obtain the set ofnumbers: 10.89%, 22.19%, 10.45%, and 6.21%. If we dividethat by a coefficient of 15, we will derive the set of 0.726%,1.479%, 0.697%, and 0.414%. Now we can compare this setwith that of the deviation to total energies from a Table 1:0.653, 1.478, 0.707, 0.406. The results of such a comparisonare very interesting, because both sets are well fitted to eachother. As it is clearly seen, the simple calculations ofdeformations lead us to the set of appropriate total energychanges with an applicable accuracy when going fromdiamond to other structural types.

And now it is possible to receive an empirical formula forrelative stability to diamond in the terms of absolute bond-angle relative deformations of the ideal tetrahedron, byanalogy to the pyramidalization angle [16] for stability ofsp2-hybridized fullerens:

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� 20

D ¼ 3:ð3Þ �X

ððP �Pd=PdÞ þ 6:ð6Þ� Sðl� ld=ldÞÞ

(1)

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where summations are performed on the six angles and fourbonds in the single deformed tetrahedron. The dimension-less quantity D for different structures is very close to theappropriate value of DE (in eV), expressing the rule of therelative stability compared to diamond structure.

The C–C bond lengths in supercubane between cubes areapprohimately 4% smaller than that of diamond, while thebond lengths in the cube itself are 2% bigger.

Considering only the cell parameters, supercubane(ac¼ 4.89 A), and tetrahedral (ac¼ 3.88 A) structures aremuch larger compared to both ‘‘C8 phase’’ polycrystallinestructures (4.28–4.29 A) and monocrystals (4.34 A). BC-8has the nearest value ac¼ 4.49 A, with applicable deviationof 3%, as reported by Johnston and Hoffmann [7]. But onlysodalite (ac¼ 4.38 A) ideally corresponds to experimentalvalues with a deviation of 1%.

Because all the structures have the same hybridizationtype, we can fit the lattice parameters and densities of theunknown phases to the experimental and calculated diamondvalues. According to these data, the deviation of latticeparameter for Matyushenko’s cubic phase of perfectstructure and for carbon sodalite equals to a value of about1%, in contrast to all other structures. The supercubanestructure is more than þ10% larger although the reflectionsare in agreement. In turn, tetragonal cubic carbon also surelycan not be considered as the crystal structure of this newphase. First of all, the difference in lattice sizes is more than�10% to the experimentally observed one and theoreticallycalculated for that atomic arrangement. Secondly, exper-imentally observed reflections of (011), (002), (112), (022),

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(013), (222) and two additional (123) (004) are also clearlyobserved in the calculated electron diffraction patterns ofsodalite (see Figure 3a). The only additional experimental(111) reflection in the single case (see Figure 1a) can beattributed to a doubled structure or other defects. Moreover,(242) and (240) reflections (see Figure 1b) are found forcalculated sodalite electron diffraction patterns.

The Liu’s C8-like crystals [8] surely can not beassociated with the supercubane structure because of itslattice parameter of approximately 4.9 A rather than reportedvalue of 5.43 A. On our opinion, (020) reflection should beconsidered as (011) one, and then, the recalculated lattice cellequals to 3.86 A. From a list of possible structures (seeTable 1) we can suppose that this phase is a tetrahedralcarbon, see Figure 3d.

4 Summary Finally, sodalite carbon structure shouldbe especially mentioned as phenomenological 3D structure.It is mathematically composed of truncated (4.6.6)-C24

octahedrons, one of the Archimedus solids. Construction ofthis crystal belongs to the old Kelvin problem of the divisionof three-dimensional space into cells of equal volume withminimal area. In 1887, he found that the truncatedoctahedron shape with uniformly partitioned space is theonly one that present the minimum surface area [17].

Sphere-like structures occupy the minimal surface forthe same volume, like Earth. Carbon 0-D analogue is knownto be fullerene C60 [18], where its spherical form became oneof the rule of stability in comparison to other fullerenes.Division of the surface on the same squares with theminimum perimeter is resolved in the nature by constructionof close packed hexagons, like honeycombs [17]. Carbon2-D analogue is known to be graphene [19]. Corresponding3D natural structures are the family of sodalite zeolites,widely used as catalysts and sieves [9]. Carbon 3D analogueshould be carbon sodalite.

As a result we resolved the Kelvin’s problem for carbonon the basis of such sodalite structure. Our calculations canhelp researchers to find the structures from a set of nearestlattice cells in the range of 3.5–5.0 A, slightly greater thandiamond, as it is shown by examples.

References

[1] E. H. L. Falcao and F. Wudl, J. Chem. Tech. Biotechnol. 82,524 (2007).

[2] A. N. Enyashin, V. G. Bamburov, and A. L. Ivanovskii, Dokl.Phys. Chem. 442, 1 (2012).

[3] V. V. Pokropivny, A. V. Pokropivny, V. V. Skorokhod, andA. V. Kurdyumov, Dopov. Natl. Akad. Nauk Ukr. 4, 112(1999).

[4] N. N. Matyushenko, V. E. Strel’nitskii, and V. A. Gusev,JETP Lett. 30, 199 (1979).

[5] L. N. Kostadinov and D. D. Dobrev, Phys. Status Solidi A109, K85 (1988).

[6] S. Fahy and S. G. Louie, Phys. Rev. B 36, 3373 (1987).[7] R. L. Johnston and R. Hoffmann, J. Am. Chem. Soc. 111, 810

(1989).

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[8] P. Liu, Y. L. Cao, C. X. Wang, X. Y. Chen, and G. W. Yang,Nano Lett. 8, 2570 (2008).

[9] B. Beagley and J. O. Titiloye, Struct. Chem. 3, 429 (1992).[10] A. A. Demkov, O. F. Sankey, J. Gryko, and P. F. McMillan,

Phys. Rev. B 55, 6904 (1997).[11] A. V. Pokropivny, Phys. Low-Dimens. Struct. 2, 64 (2006).[12] V. A. Greshnyakov and E. A. Belenkov, JETP 113, 86 (2011).[13] A. N. Enyashin and A. L. Ivanovsky, Phys. Status Solidi B

248, 1879 (2011).[14] E. Artacho, E. Anglada, O. Dieguez, J. D. Gale, A. Garcia,

J. Junquera, R. M. Martin, P. Ordejon, J. M. Pruneda,

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D. Sanchez-Portal, and J. M. Soler, J. Phys.: Condens. Matter20, 064208 (2008).

[15] S. Fleming and A. Rohl, Z. Kristallogr. 220, 580 (2005).[16] T. C. Dinadayalane and J. Leszczynski, Struct. Chem. 21,

1155 (2010).[17] P. Pearce, Structure in nature is a strategy for design (MIT

Press, 1990).[18] W. Kratschmer and D. R. Huffman, Philos. Trans. R. Soc.

Lond. A 343, 33 (1993).[19] A. K. Geim and K. S. Novoselov, Nature Mater. 6, 183

(2007).

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The prices listed below are valid only for orders received in the course of 20 .

Single issues are available to authors at a reduced price.All prices are subject to local VAT/sales tax.

All reprints are delivered with color cover

€

Special offer: If you order 100 or more re-prints you will receive a pdf file (300 dpi,unlimited number of printouts, color figures) and an issue for free.

Please note that for German sales tax purposes the charge for color print is considered a service and therefore is subject to German ales tax. For institutional customers in other countries the tax will be waived, i.e. the Recipient ofService is liable for VAT. Members of the EU will have to present a VAT identification number. Customers in other countries may also be asked to provide according tax identification information.

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