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The Seven Crystal Systems: Overview01:24

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...
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Structures of Solids

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...
Crystallographic Point Groups01:29

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...
Crystal Field Theory - Tetrahedral and Square Planar Complexes02:46

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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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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...
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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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Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses
08:55

Methods of Ex Situ and In Situ Investigations of Structural Transformations: The Case of Crystallization of Metallic Glasses

Published on: June 7, 2018

Rigid-unit modes in tetrahedral crystals.

Franz Wegner1

  • 1Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 19, D-69120 Heidelberg, Germany.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|November 4, 2011
PubMed
Summary
This summary is machine-generated.

The rigid-unit mode (RUM) model reveals how SiO(4) tetrahedra vibrations form surfaces or lines of RUMs in crystal lattices. This depends on the crystal

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Area of Science:

  • Solid-state physics
  • Crystallography
  • Materials science

Background:

  • The rigid-unit mode (RUM) model simplifies lattice dynamics by treating structural units as rigid.
  • Understanding lattice vibrations is crucial for predicting material properties.

Purpose of the Study:

  • To analytically determine the wavevectors of lattice vibrations obeying the RUM model for SiO(2) polymorphs.
  • To investigate how crystal symmetry, specifically inversion symmetry, influences the occurrence of RUMs in reciprocal space.

Main Methods:

  • Analytical determination of wavevectors for lattice vibrations under the RUM constraint.
  • Application of the RUM model to tetrahedra of SiO(4) groups as rigid units.
  • Examination of five different SiO(2) crystal polymorphs.

Main Results:

  • Lattices with inversion symmetry generically exhibit surfaces of RUMs in reciprocal space.
  • Lattices lacking inversion symmetry typically display lines of RUMs.
  • Exceptional cases, like β-quartz, can show surfaces of RUMs even without inversion symmetry.

Conclusions:

  • The symmetry of a crystal lattice fundamentally dictates the geometric manifestation of rigid-unit modes in reciprocal space.
  • The RUM model provides a framework for understanding specific vibrational modes in silica polymorphs.
  • Further analysis of RUMs can elucidate structure-property relationships in crystalline materials.