On an Exact Mapping and a Higher Order Born Rule for Use in Analyzing Graphene Carbon Nanotubes

Report No. ARL-TR-3117
Authors: Michael J. Leamy, Peter W. Chung, and Raju Namburu
Date/Pages: December 2003; 24 pages
Abstract: As part of an effort to model single-walled graphene carbon nanotubes, a transformation is first described which maps atom locations originally on a planar sheet to atom locations on a cylindrical nanotube. The mapping is parametrized by the two components describing the chiral vector, and thus is valid for generating the atomic locations of arbitrary chirality nanotubes. Attention is then turned towards quantifying the dimensions of the unit cell, again parametrized by the chiral vector?s two components. Taken together, the mapping plus the unit cell dimensions are used to generate unique atomic locations of arbitrarily long carbon nanotubes. A second mapping is described which generalizes the commonly used Born rule to higher orders. The Born rule is an effective tool for linking bulk material deformations to atomic displacements, particularly for homogenous deformations. For nonhomogenous deformations, such as those involving nonzero curvature, the standard Born rule can inaccurately link the two scales. The higher order mapping described herein allows nonhomogenous deformations to be mapped down to the atomic scale to an arbitrary degree of precision.
Distribution: Approved for public release
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Last Update / Reviewed: December 1, 2003