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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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Metallic Solids02:37

Metallic Solids

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Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
All metallic solids exhibit high thermal and electrical conductivity, metallic luster, and...
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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
Imagine taking a large number of identical...
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Ionic Crystal Structures02:42

Ionic Crystal Structures

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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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Coordination Number and Geometry02:57

Coordination Number and Geometry

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For transition metal complexes, the coordination number determines the geometry around the central metal ion. Table 1 compares coordination numbers to molecular geometry. The most common structures of the complexes in coordination compounds are octahedral, tetrahedral, and square planar.
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Network Covalent Solids02:18

Network Covalent Solids

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Network covalent solids contain a three-dimensional network of covalently bonded atoms as found in the crystal structures of nonmetals like diamond, graphite, silicon, and some covalent compounds, such as silicon dioxide (sand) and silicon carbide (carborundum, the abrasive on sandpaper). Many minerals have networks of covalent bonds.
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Geometric structure of percolation clusters.

Xiao Xu1, Junfeng Wang1, Zongzheng Zhou2

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Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary
This summary is machine-generated.

Percolation clusters on a torus exhibit specific geometric properties. Bridges and nonbridges stabilize at 1/4 density, with branches and junctions influencing fractal dimensions in modified configurations.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Geometric Properties of Matter

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • Understanding the geometric properties of these clusters is crucial for various scientific disciplines.
  • Previous research has focused on general cluster characteristics, but specific bond classifications require further investigation.

Purpose of the Study:

  • To investigate the geometric properties of percolation clusters on a square lattice torus.
  • To classify and analyze the roles of bridges and nonbridges within these clusters.
  • To determine the impact of removing specific bond types on cluster geometry and fractal dimensions.

Main Methods:

  • Utilized square-lattice bond percolation on a torus for simulations.
  • Employed Monte Carlo simulations to analyze bond properties and probabilities.
  • Classified bridges into 'branches' and 'junctions' based on cluster topology after deletion.
  • Calculated fractal dimensions for cluster size and hull length in modified configurations.

Main Results:

  • The density of both bridges and nonbridges approaches 1/4 for large system sizes.
  • Identified a bridge as a 'branch' if its removal leaves at least one tree cluster.
  • Leaf-free configurations (after branch removal) maintain standard percolation fractal dimensions.
  • Bridge-free configurations exhibit fractal dimensions related to the backbone and external perimeter, with an estimated backbone fractal dimension of 1.64336(10).

Conclusions:

  • The study provides a detailed geometric characterization of percolation clusters by analyzing bond types.
  • The classification of bridges into branches and junctions offers new insights into cluster structure.
  • Removing branches preserves standard fractal properties, while removing all bridges reveals distinct backbone and perimeter scaling.