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Algebraic determination of generating functions for coordination sequences in crystal structures.

Jean Guillaume Eon1

  • 1Instituto de Química, Universidade Federal do Rio de Janeiro, A-632, Cidade Universitária, 21945-970 Ilha do Fundão, Rio de Janeiro, Brazil. jgeon@iq.ufrj.br

Acta Crystallographica. Section A, Foundations of Crystallography
|December 26, 2001
PubMed
Summary

This study embeds crystalline structure paths into a commutative ring, using algebraic methods to generate network structures and their coordination sequences.

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Area of Science:

  • Materials Science
  • Graph Theory
  • Algebraic Topology

Background:

  • Crystalline structures possess inherent network properties.
  • Understanding paths and cycles within these structures is crucial for materials design.
  • Existing methods lack a unified algebraic framework for analyzing network topology.

Purpose of the Study:

  • To embed the paths and cycles of a crystalline structure's bond net into a commutative ring.
  • To develop an algebraic generator for network structures based on topological criteria.
  • To establish a ring mapping for generating the coordination sequence.

Main Methods:

  • Embedding paths and cycles of the quotient graph into a commutative ring.
  • Utilizing multiplication for combining walks and addition for enumerating walks.

Related Experiment Videos

  • Applying topological criteria to define zero divisors.
  • Developing an algebraic generator for geodesics.
  • Establishing a ring mapping to derive the generating function.

    • Main Results:

    • A commutative ring structure is established for crystalline structure nets.
    • Zero divisors are topologically defined, enabling algebraic generation.
    • An algebraic generator is developed, corresponding to net vertices.
    • A ring mapping successfully yields the generating function for coordination sequences.

    Conclusions:

    • The developed commutative ring provides a powerful algebraic framework for analyzing crystalline structures.
    • The algebraic generator offers a novel method for constructing network components.
    • The generating function derived from ring mapping accurately represents coordination sequences.