Topo-combinatoric categorization of quasi-local graphitic defects

A quasi-local graphitic defect is defined as finite network surrounded by a trivalent planar hexagonal-faced network extending to infinity. A general classification of such quasi-local defects is developed, so as to attend to infinite-range structural consequences. A first classification stage identifies a discrete “combinatorial curvature” characteristic κ, associating closely to geometric Gaussian curvature of embeddings of the network surface in Euclidean space. If κ = 2π, a singly capped bucky-tube results, whereas if 0 < κ < 2π, it is a positive-curvature graphitic cone. If κ < 0, a “fluted” or “crenelated” cone results, and if κ = 0, it is a globally “flat” (in a Euclidean sense) structure. A second stage of classification identifies a topo-combinatoric “circum-matching” characteristic. For the graphitic-cone case there may be more than one “circum-matching” class for each κ, so that 8 such classes arise for the 5 allowed values of κ, with 0 < κ < 2π. A similar result applies for the negatively curved cones with −2π < κ < 0. On the other hand, there is an infinity of classes for the “flat” κ = 0 case, where each circum-matching class corresponds to a standard Burgers dislocation vector. There are also an infinity of classes for κ = 2π or for κ any positive integer multiple of −2π. A further “quasi-spin” characteristic refines the classes of quasi-local defects into “irrotational” subclasses, as are relevant for multi-wall cones. Combining rules for pairs of defects are considered.