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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
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Crystals with various point group symmetries belong to different crystal classes, which are synonymous terms. Despite being in the same class, crystals may have distinct shapes, like cubes and octahedra. There are 32 three-dimensional point groups, all of which are systematically divided into seven crystal systems.The basic cubic crystal system, exemplified by NaCl, features orthogonal vectors (α = β = �� = 90°) of equal lengths (a = b = c). When specific...
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Crystallographic point groups represent the various symmetry operations that can occur within crystals. They are unique in that at least one point will always remain unchanged during these actions. For instance, consider the triclinic system. This system, devoid of any axis or plane of symmetry, aligns with the C1 and Ci point groups.where Cᵢ is characterized solely by a center of inversion.Contrastingly, the monoclinic system introduces an element of symmetry. This system with one plane...
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A crystal's internal structure is an orderly array of atoms, ions, or molecules, and the details of this array significantly influence the solid's properties. In a crystal, periodically repeating 'structural motifs' - which could be atoms, molecules, or groups thereof - create a 'space lattice.' This is essentially a three-dimensional, infinite array of points, each surrounded by its neighbors in an identical way, forming the basic structure of the crystal.A 'unit cell' is a theoretical...
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Sample Preparation and Transfer Protocol for In-Vacuum Long-Wavelength Crystallography on Beamline I23 at Diamond Light Source
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A topological coordinate system for the diamond cubic grid.

Lidija Čomić1, Benedek Nagy2

  • 1Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia.

Acta Crystallographica. Section A, Foundations and Advances
|September 1, 2016
PubMed
Summary

A new topological coordinate system addresses cells in the diamond cubic grid. This system simplifies capturing cell relations for applications like shape analysis and visualization.

Keywords:
abstract cell complexesdiamond cubic gridnon-traditional three-dimensional gridstopological coordinate systemtopological operations

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Area of Science:

  • Computational geometry
  • Crystallography
  • Topology

Background:

  • Topological coordinate systems are essential for addressing cells in abstract cell complexes.
  • The diamond cubic grid is a fundamental structure in materials science and crystallography.

Purpose of the Study:

  • To present a novel topological coordinate system for the diamond cubic grid.
  • To detail the properties and applications of this coordinate system.

Main Methods:

  • Development of a coordinate system using four dependent coordinates.
  • Addressing voxels, faces, edges, and corner points within the diamond cubic grid.

Main Results:

  • The proposed system effectively addresses all cell types in the diamond cubic grid.
  • Boundary, co-boundary, and adjacency relations are easily captured by coordinate values.

Conclusions:

  • The developed topological coordinate system is suitable for the diamond cubic grid.
  • It facilitates implementation in diverse applications including visualization, morphological operations, and shape analysis.