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Lattice Centering and Coordination Number02:33

Lattice Centering and Coordination Number

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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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Unit Cells01:18

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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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Structures of Solids02:22

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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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The Seven Crystal Systems: Overview01:24

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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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In classical mechanics, the two-body problem is one of the fundamental problems describing the motion of two interacting bodies under gravity or any other central force. When considering the motion of two bodies, one of the most important concepts is the reduced mass coordinates, a quantity that allows the two-body problem to be solved like a single-body problem. In these circumstances, it is assumed that a single body with reduced mass revolves around another body fixed in a position with an...
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Crystallographic Point Groups01:29

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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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The Normalized Reduced Form and Cell Mathematical Tools for Lattice Analysis-Symmetry and Similarity.

Alan D Mighell1

  • 1National Institute of Standards and Technology, Gaithersburg, MD 20899-8520.

Journal of Research of the National Institute of Standards and Technology
|July 15, 2016
PubMed
Summary

Mathematical tools like the normalized reduced form and normalized reduced cell help analyze crystallographic databases. These methods reveal lattice relationships, aiding structure solving, materials design, and nanotechnology applications.

Keywords:
identificationlattice relationshipslattice similaritylattice-matching strategiesmetric latticenormalized reduced cell and formsymmetry

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Area of Science:

  • Crystallography and Materials Science

Background:

  • Effective use of crystallographic databases requires advanced mathematical and computational tools.
  • Understanding intra- and interlattice relationships is crucial for data analysis.

Purpose of the Study:

  • To introduce and highlight the utility of the normalized reduced form and normalized reduced cell.
  • To demonstrate their application in analyzing crystallographic data and identifying similar lattices.

Main Methods:

  • Utilizing the normalized reduced form to establish lattice metric symmetry.
  • Employing the normalized reduced cell for determining metrically similar lattices.

Main Results:

  • The normalized reduced form simplifies the deduction of relationships within reduced forms.
  • The normalized reduced cell is vital for identifying lattices with metric similarity.

Conclusions:

  • Normalized reduced cell and form are essential tools for crystallographic database utilization.
  • These methods support structure solving, materials design, and nanotechnology.
  • Routine application in database searching, data evaluation, and experimental work is recommended.