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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.
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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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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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Interdependent lattice networks in high dimensions.

Steven Lowinger1, Gabriel A Cwilich1, Sergey V Buldyrev1

  • 1Department of Physics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA.

Physical Review. E
|December 15, 2016
PubMed
Summary
This summary is machine-generated.

We studied interdependent lattice networks and found that link distance influences failure cascades. Shorter link distances can lead to continuous transitions, while longer distances may cause abrupt collapses in these complex systems.

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Area of Science:

  • Statistical Physics
  • Network Science
  • Complex Systems

Background:

  • Interdependent networks are crucial in real-world systems.
  • Percolation theory studies network robustness and failure dynamics.
  • Understanding cascading failures is vital for system resilience.

Purpose of the Study:

  • To investigate mutual percolation in two interdependent lattice networks.
  • To analyze the impact of interdependency link length on system robustness.
  • To characterize the nature of phase transitions in these systems.

Main Methods:

  • Simulating mutual percolation on D-dimensional lattices (2 ≤ D ≤ 7).
  • Varying the chemical distance (r) of interdependency links.
  • Analyzing the mutual percolation threshold (p_c[D,r]) and transition order.

Main Results:

  • A critical link distance (r_I) determines transition order: discontinuous for r ≥ r_I, continuous for r < r_I (for D < 6).
  • For D=6, r_I=1, matching random regular graphs.
  • Maximal system vulnerability occurs at r_max, with decreasing vulnerability for r > r_max, especially at higher dimensions.

Conclusions:

  • Link distance significantly alters the failure dynamics and transition order in interdependent networks.
  • Higher dimensions stabilize the system against cascading failures.
  • The findings have implications for designing robust interdependent systems.