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Related Concept Videos

Structures of Solids02:22

Structures of Solids

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Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
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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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Crystal Field Theory - Octahedral Complexes02:58

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Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
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Lattice Centering and Coordination Number02:33

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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.
Types of Unit Cells
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X-ray Crystallography02:18

X-ray Crystallography

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The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography.
Diffraction
Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring...
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Polymer Classification: Crystallinity01:21

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Unlike ionic or small covalent molecules, polymers do not form crystalline solids due to the diffusion limitations of their long-chain structures. However, polymers contain microscopic crystalline domains separated by amorphous domains.
Crystalline domains are the regions where polymer chains are aligned in an orderly manner and held together in proximity by intermolecular forces. For example, chains in the crystalline domains of polyethylene and nylon are bound together by van der Waals...
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Related Experiment Video

Updated: Aug 10, 2025

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
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Classification of time-reversal-invariant crystals with gauge structures.

Z Y Chen1, Zheng Zhang1, Shengyuan A Yang2

  • 1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, 210093, Nanjing, China.

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|February 10, 2023
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Summary
This summary is machine-generated.

This study unifies the theory of projective representations for crystal symmetries, classifying 458 algebras. It reveals three physical signatures, enabling new topological states in artificial crystals.

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Area of Science:

  • Condensed Matter Physics
  • Quantum Mechanics
  • Crystallography

Background:

  • Quantum states can exhibit projective representations of symmetries, crucial for understanding crystals.
  • Projective representations of space groups are vital for artificial crystals but lack a unified theory.

Purpose of the Study:

  • To establish a unified theory for projective symmetry algebras in time-reversal-invariant crystals.
  • To classify and represent all projective symmetry algebras for 17 wallpaper groups in 2D.

Main Methods:

  • Exhaustive classification of 458 projective symmetry algebras.
  • Representation of these algebras for 2D time-reversal-invariant crystals.

Main Results:

  • Identified 189 algebraically non-equivalent projective symmetry algebras.
  • Discovered three physical signatures: shifted high-symmetry momenta, nontrivial Zak phase, and a spinless eight-fold nodal point.

Conclusions:

  • Provides a theoretical foundation for artificial crystals and projective symmetry.
  • Opens avenues for novel topological states and phenomena beyond current paradigms.