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Geographic style maps for two-dimensional lattices.

Matthew Bright1, Andrew I Cooper1, Vitaliy Kurlin1

  • 1Materials Innovation Factory, University of Liverpool, UK.

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|January 5, 2023
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Summary
This summary is machine-generated.

This study introduces geographic style maps for crystal lattices, visualizing all periodic crystal structures. These maps offer a new continuous framework for crystallography, aiding in the analysis of crystal structures.

Keywords:
complete invariantscontinuityisometrymetric tensorobtuse superbasereduced basistwo-dimensional lattices

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Area of Science:

  • Crystallography
  • Materials Science
  • Geometry

Background:

  • Periodic crystals are fundamental to materials science.
  • Understanding the geometric relationships between crystal lattices is crucial.
  • Existing methods for classifying and visualizing lattices have limitations.

Purpose of the Study:

  • To develop a novel geographic style mapping system for two-dimensional lattices.
  • To parameterize all known periodic crystals using complete invariants.
  • To visualize and analyze the space of two-dimensional lattices up to rigid motion and uniform scaling.

Main Methods:

  • Development of continuous maps representing the space of two-dimensional lattices.
  • Parameterization of lattices using recent complete invariants.
  • Utilizing the Cambridge Structural Database for real crystal structure data.

Main Results:

  • The space of two-dimensional lattices is characterized as a square with identified edges or a punctured sphere.
  • All Bravais classes are represented as low-dimensional subspaces within the continuous maps.
  • Visualization of hundreds of thousands of real crystal structures from the Cambridge Structural Database.

Conclusions:

  • The developed maps provide a continuous and invariant-based framework for crystallography.
  • This approach offers new insights into the geometric properties of crystal structures.
  • The methodology motivates advancements in continuous and invariant-based crystallographic analysis.