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Related Concept Videos

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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Trends in Lattice Energy: Ion Size and Charge02:54

Trends in Lattice Energy: Ion Size and Charge

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An ionic compound is stable because of the electrostatic attraction between its positive and negative ions. The lattice energy of a compound is a measure of the strength of this attraction. The lattice energy (ΔHlattice) of an ionic compound is defined as the energy required to separate one mole of the solid into its component gaseous ions. For the ionic solid sodium chloride, the lattice energy is the enthalpy change of the process:
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Bewley Lattice Diagram01:12

Bewley Lattice Diagram

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The Bewley lattice diagram, developed by L. V. Bewley, effectively organizes the reflections occurring during transmission-line transients. It visually represents how voltage waves propagate and reflect within a transmission line, making it easier to understand the complex interactions that occur.
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Density00:56

Density

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Density is an important characteristic of substances, crucial in determining whether an object sinks or floats in a fluid. Its SI unit is kg/m3, and its cgs unit is g/cm3. The density of an object helps in identifying its composition, and also reveals information about the phase of the matter and its substructure. The densities of liquids and solids are roughly comparable, consistent with the fact that their atoms are in close contact. However, gases have much lower densities than liquids and...
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Current Density01:21

Current Density

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The total amount of current flowing through one unit value of a cross-sectional area is referred to as current density. If the current flow is uniform, the amount of current flowing through a conductor is the same at all points along the conductor, even if the conductor area varies. The current density consists of the local magnitude and direction of the charge flow, which varies from point to point. Current density is measured in amperes per meter square, and direction is defined as the net...
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Bulk Density of Aggregate01:22

Bulk Density of Aggregate

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Bulk density refers to the mass of aggregate particles that would fill a unit volume. The concept of bulk density originates from the inability to pack aggregate particles in a manner that completely eliminates void spaces. Hence, the term bulk refers to the volume that encompasses both the aggregates and the voids. This measurement is crucial when aggregates are batched by volume and is used to convert quantities by mass to volume.
Most natural mineral aggregates, like sand and gravel,...
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Related Experiment Video

Updated: Jan 23, 2026

Preparation of Synaptoneurosomes from Mouse Cortex using a Discontinuous Percoll-Sucrose Density Gradient
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Preparation of Synaptoneurosomes from Mouse Cortex using a Discontinuous Percoll-Sucrose Density Gradient

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High-density percolation on the modified Bethe lattice.

C Widder1, T Schilling1

  • 1Physikalisches Institut Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Strasse 3, 79104 Freiburg, Germany.

Physical Review. E
|June 20, 2019
PubMed
Summary

High-density percolation on modified Bethe lattices was analyzed using random graph theory. The study derived key percolation properties and found critical exponents beta and gamma equal to 1.

Area of Science:

  • Statistical Physics
  • Network Science
  • Graph Theory

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • High-density percolation specifically examines cluster formation with a minimum number of occupied neighbors.
  • Modified Bethe lattices and large random graphs provide frameworks for analyzing complex network structures.

Purpose of the Study:

  • To investigate high-density percolation phenomena on modified Bethe lattices.
  • To apply the theory of large random graphs with arbitrary degree distributions to this percolation model.
  • To derive and analyze key percolation parameters including cluster size distribution, threshold, and critical exponents.

Main Methods:

  • Utilized the formalism of generating functions to derive analytical expressions.

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  • Applied concepts from the theory of large random graphs.
  • Employed numerical solutions and simulation methods for comparison.

    • Main Results:

    • Derived expressions for cluster size distribution, percolation threshold, percolation probability, and mean size of finite clusters.
    • Determined that the critical exponents beta and gamma are both equal to 1.
    • Validated theoretical findings through comparison with numerical and simulation data.

    Conclusions:

    • The study successfully characterized high-density percolation on modified Bethe lattices using random graph theory.
    • The derived critical exponents (β=γ=1) provide fundamental insights into the universality of percolation phenomena in these networks.
    • The findings are supported by consistent numerical and simulation results, enhancing confidence in the theoretical framework.