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Topological density of nets: a direct calculation.

Jean Guillaume Eon1

  • 1Instituto de Química, Universidade Federal do Rio de Janeiro, A-631 Cidade Universitária, Ilha do Fundão, Rio de Janeiro, 21945-970 Brazil. jgeon@iq.ufrj.br

Acta Crystallographica. Section A, Foundations of Crystallography
|December 24, 2003
PubMed
Summary

Calculating the topological density of 3D nets is simplified. This study introduces a new method using the cycles of a net

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Area of Science:

  • Materials Science
  • Crystallography
  • Network Theory

Background:

  • Topological density calculation for 3D nets traditionally requires determining the full coordination sequence.
  • This process is often complex and computationally intensive.

Purpose of the Study:

  • To introduce a novel, direct method for calculating the topological density of three-dimensional nets.
  • To develop a geometrical tool, the 'cycles figure', for this calculation.
  • To extend the method to n-dimensional nets.

Main Methods:

  • Utilizing the set of cycles from the quotient graph of a net.
  • Developing and describing the construction of the 'cycles figure' as a convex polyhedron with triangular faces.
  • Deriving an exact mathematical expression for topological density.

Main Results:

  • A direct method for computing topological density from the net's cycles is established.
  • The 'cycles figure' is presented as an effective geometrical tool for this calculation.
  • An exact expression for topological density is derived for 3D nets and generalized to n-dimensions.

Conclusions:

  • The 'cycles figure' method offers a more direct and efficient approach to determining net topological density.
  • The derived exact expression simplifies calculations for various dimensional nets.
  • Applications are demonstrated for 3D lattices, nets, and n-dimensional diamond and lonsdaleite structures.

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