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GraphT-T (V1.0Beta), a program for embedding and visualizing periodic graphs in 3D Euclidean space.

Maxwell Christopher Day1, Ali Rostami2, Frank Christopher Hawthorne1

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Summary

The GraphT-T program models silicate chains, determining if their structures are compatible with crystal formation. It uses geometric constraints based on silicon-oxygen tetrahedra distances to predict crystal structure viability.

Keywords:
(SiO4)4− tetrahedra3D Euclidean space3D spring-force algorithmGraphT–Tbond topologychains of tetrahedragraph embedding program

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Area of Science:

  • Crystallography
  • Materials Science
  • Computational Chemistry

Background:

  • Silicate minerals are fundamental to Earth's crust, with their structures determined by chains of silicon-oxygen tetrahedra.
  • Understanding the geometric constraints governing these tetrahedral arrangements is crucial for predicting mineral formation and properties.
  • Previous work established foundational principles for analyzing silicate chain structures in Euclidean space.

Purpose of the Study:

  • To develop and apply the GraphT-T program for embedding silicate tetrahedral chains into 2D and 3D space.
  • To establish criteria for determining the compatibility of hypothetical and observed silicate chain graphs with crystal structures.
  • To investigate the influence of topological properties on the flexibility and occurrence of silicate chain arrangements in minerals.

Main Methods:

  • Developed the GraphT-T program to represent chains of (SiO4)4- tetrahedra as graphs in Euclidean space.
  • Employed a 3D spring-force algorithm simulating Hooke's law for linked vertices and Coulomb's law for unlinked vertices.
  • Iteratively refined embedding parameters (spring coefficient k, Coulomb's constant K) to match observed silicate mineral distances (T-T: 3.06±0.15 Å, T...T: ≥3.713 Å).

Main Results:

  • Defined 'compatible' unit-distance graphs that satisfy geometric constraints (T-T and T...T distances) observed in silicate minerals.
  • Identified 'incompatible' graphs that do not meet these geometric criteria, suggesting they are unlikely to form crystal structures.
  • Demonstrated that topological properties of chain graphs significantly influence the flexibility and commonality of specific silicate arrangements in minerals.

Conclusions:

  • The GraphT-T program provides a robust method for assessing the crystallographic potential of silicate chain structures.
  • Geometric compatibility, defined by specific inter-tetrahedral distances, is a key factor determining whether a silicate chain can form a crystal structure.
  • The topological characteristics of silicate chains explain the observed prevalence of certain mineral structures and the absence of others.