ࡱ> 9 #bjbj ,Blx x x 8  $d  (: : : M M M GIII5~R&$9 YtJEM I M M M J: : M : : GM G33: @ [x 3303e3b^ Lithographite: A 5-,6-Connected Orthorhombic Structural Pattern in Space Group Pmmm Michael J. Bucknum Georgia College & State University, Department of Chemistry and Physics, CBX 82, Milledgeville, GA 31061, U.S.A. Eduardo A. Castro* INIFTA, Theoretical Chemistry Division, Suc.4, C.C. 16, La Plata 1900, Argentina Abstract-This paper describes the crystal structure of a novel, hypothetical 5-,6-connected orthorhombic structural pattern with the binary stoichiometry AB2. The novel pattern lies in space group Pmmm, number 47. It possesses the Wells point symbol (41062)(4762)2 and has the Wellsean Schlfli index (42/5, 51/3). The pattern represents a fusing together of what are termed lithographene sheets. Such sheets can be envisioned as a projection of the lithographite lattice in (001), they represent a tiling of the Euclidean plane with hexagons and rhombi, where the rhombi have acute and obtuse angles of 60 and 120, respectively. The lithographene sheet, which occurs in plane group p2mm, is compared to the ordinary graphene sheet in plane group p6mm. * Corresponding author ( HYPERLINK "mailto:castro@quimica.unlp.edu.ar" castro@quimica.unlp.edu.ar / direccion@inifta.unlp.edu.ar) 1. Introduction In the course of investigating hypothetical structural modifications in the orthorhombic symmetry class, [1] one of the authors (MJB) discovered a novel 5-,6-connected structure-type with a binary stoichiometry AB2. The new structure-type is named lithographite. An idealized illustration of the unit of pattern of this new structure-type, shown in plane projection in (001), is shown in Figure 1. Figure 2 shows an extended view of the lattice in plane projection in (001), such a pattern will be called a lithographene substructure of the layered lithographite lattice. Figure 1 Figure 2 The lattice has been given the name lithographite because it is largely seen as a layering of a 2-dimensional lithographic tiling of the Euclidean plane by hexagons and rhombi. The dominant appearance of hexagons in this lithographene pattern is reminiscent of the much more symmetrical graphene sheet from which its name is derived. 2. Lithographite Referring to the unit cell of lithographite shown in plane projection in (001) in Figure 1, the white circles represent 6-connected distorted octahedral atoms, there is one such white circle, labeled A, per unit cell; while the black circles represent the 5-connected undistorted trigonal bipyramidal atoms, there are two such black circles, labeled B, per unit cell. This orthorhombic 5-,6-connected structural pattern, with binary stoichiometry AB2, lies in space group Pmmm, number 47, with atoms in the following Wyckoff positions: the one 6-connected octahedral atom is in 1(a) (0, 0, 0), and the two 5-connected trigonal bipyramidal atoms are in 2(k) (x, 1/2, 0). For the special symmetry in which the angle a in Figure 1 is set exactly to 120, the rhombi are of acute and obtuse angles of 60 and 120, respectively. [2] The topology of the crystalline materials has been developed elsewhere and does not require a separate description here. [3] The lithographite lattice possesses the Wells point symbol (41062)(4762)2 and has the Wellsean Schlfli index (42/5, 51/3). 3. Lithographene Sheets Referring to the extended drawing of lithographite shown in plane projection in (001) in Figure 2, one can see that this sheet by itself represents a, by now, familiar tiling of the Euclidean plane with hexagons in strips that are so aligned with parallel hexagons in strips that rhombi are inscribed in the areas where the fused hexagons have frustrations. If we set the unit cell parameter a equal to 120, the internal hexagonal angles are ideal at 120. This lithographene pattern is a 3-,4-connected plane net. By removing all of the neighboring lithographene layers in lithographite away from each other, one has transformed the octahedral vertices in lithographite into distorted square planar 4-connected vertices in the lithographene sheet, by truncation. Similarly, the trigonal bipyramidal vertices of lithographite have been transformed into ideal trigonal planar 3-connected vertices in the lithographene sheet. The Wells point symbol for the sheet is given by (4262)(462)2. One can see immediately that this is the point symbol for a topologically irregular pattern, its Schlfli index is (51/5, 31/3). In comparison to this topology, one can consider the graphene sheet with its regular (Platonic) topology of (6, 3). The construction of the lithographene sheet from the graphene sheet can easily be envisioned, the latter sheet is shown in Figure 3. Figure 3 In both structures one has strips of hexagons which are fused into each other. The difference is in the alignment of the hexagon strips, for in graphene the strips are completely regularly connected into each other leading to a high topology, a high plane group symmetry of p6mm and the absence of any frustrations. [4] This is achieved by aligning the hexagon strips such that hexagons from one strip are related to hexagons from the next-to-nearest strips by a reflection plane, m. In other words, every other strip of parallel hexagons is related by a mirror plane through the intervening strip of hexagons. In contrast, in the lithographene sheet the topology is irregular, the symmetry of its plane group is lowered to p2mm and the alignment of its hexagon strips leads to the introduction of rhomboid frustrations. The great contrast can be understood in terms of the alignment of the strips of hexagons in the latter situation, for in lithographene each and every parallel hexagon strip is related to every other one, including its adjacent neighbors, by reflection symmetry planes, m, passing between and also through the hexagon strips. [4] 4. Acknowledgement One of us (M. J. B.) is grateful for the support of his wife, Hsi-cheng Shen. He is also grateful to Jane Jorgensen and Ali Avcisoy for preparing the structural drawings of the lithographite lattice. References 1. (a) A.F. Wells, Structural Inorganic Chemistry, 4th edition, Oxford University Press, Oxford, U.K., 1984. (b) M. OKeeffe and B.G. Hyde, Crystal Structures I. Patterns and Symmetry, Mineralogical Society of America, Washington, D.C., 1996. 2. T. Hahn, editor, International Tables for Crystallography, Volume A: Space Group Symmetry, 4th edition, Kluwer Academic Publishers, Dordrecht/ Boston/ London, 1995. 3. (a) Elementa doctrinae solidorum and Demonstratio nonnularum insignium proprietatum quibus solida heddris planis inclusa sunt praedita, L. Euler, included in the Proceedings of the St. Petersburg Academy, 1758. (b) A.F. Wells, Three Dimensional Nets and Polyhedra, 1st edition, John Wiley and Sons Inc., New York, 1977. (c) M.J. Bucknum, Carbon, 35, 1, (1997). (d) M. Henle, A Combinatorial Introduction to Topology, 1st Edition, W.H. Freeman and Company, San Francisco, CA, (1979), p. 9. (e) "Calculating Topological Indexes for a Network Based Upon the Wells Point Symbol for the Network", M.J. Bucknum, in preparation. 4. C. Hammond, Royal Microscopical Society Microscopy Handbooks (19): Introduction to Crystallography, Revised Edition, Oxford University Press, Oxford, U.K., 1992, p.29.  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