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Crystal Field Theory - Tetrahedral and Square Planar Complexes02:46

Crystal Field Theory - Tetrahedral and Square Planar Complexes

Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
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Crystallographic Point Groups

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 and...
Lattice Centering and Coordination Number02:33

Lattice Centering and Coordination Number

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.
Types of Unit Cells
Imagine taking a large number of identical...
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The Seven Crystal Systems: Overview

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 requirements are not imposed on the...
Unit Cells01:18

Unit Cells

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...
Ionic Crystal Structures02:42

Ionic Crystal Structures

Ionic crystals consist of two or more different kinds of ions that usually have different sizes. The packing of these ions into a crystal structure is more complex than the packing of metal atoms that are the same size.
Most monatomic ions behave as charged spheres, and their attraction for ions of opposite charge is the same in every direction. Consequently, stable structures for ionic compounds result (1) when ions of one charge are surrounded by as many ions as possible of the opposite...

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Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles
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Point substitution processes for decagonal quasiperiodic tilings.

Nobuhisa Fujita1

  • 1Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan. nobuhisa@tagen.tohoku.ac.jp

Acta Crystallographica. Section A, Foundations of Crystallography
|August 19, 2009
PubMed
Summary
This summary is machine-generated.

A new method generates decagonal quasiperiodic tilings using unit-edged polygons. This approach reveals novel tiling families with fractal boundaries, expanding the understanding of quasicrystal structures.

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Area of Science:

  • Materials Science
  • Crystallography
  • Mathematical Physics

Background:

  • Quasiperiodic tilings, particularly decagonal ones, exhibit complex structures.
  • Understanding their construction principles is key to exploring their properties.

Purpose of the Study:

  • To propose a general construction principle for inflation rules of decagonal quasiperiodic tilings.
  • To introduce new families of decagonal tilings and investigate their properties.

Main Methods:

  • Defining inflation rules as tile expansion and division, maintaining prototile sets.
  • Utilizing a point decoration process to identify valid division rules.
  • Investigating properties of ternary tilings with specific prototiles.

Main Results:

  • A general construction principle for decagonal tiling inflation rules is established.
  • Two new families of decagonal tilings (quaternary and ternary) are generated.
  • Many generated tilings are chiral and possess fractal atomic surfaces.

Conclusions:

  • The proposed method offers a versatile approach to generating diverse decagonal tilings.
  • The study details properties of ternary tilings, contributing to quasicrystal research.