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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.
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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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Valence Bond Theory

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Coordination compounds and complexes exhibit different colors, geometries, and magnetic behavior, depending on the metal atom/ion and ligands from which they are composed. In an attempt to explain the bonding and structure of coordination complexes, Linus Pauling proposed the valence bond theory, or VBT, using the concepts of hybridization and the overlapping of the atomic orbitals. According to VBT, the central metal atom or ion (Lewis acid) hybridizes to provide empty orbitals of suitable...
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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.
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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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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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Three-state Potts model on the centered triangular lattice.

Zhe Fu1, Wenan Guo2, Henk W J Blöte3

  • 1College of Physics and Electronic Engineering, Xinxiang University, Xinxiang 453003, China.

Physical Review. E
|February 20, 2020
PubMed
Summary

This study investigates the Potts model on a centered-triangular lattice, revealing two phase transitions in the antiferromagnetic regime. These transitions exhibit unique critical properties, especially in the geometric frustration cases.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics

Background:

  • The Potts model is a fundamental tool for studying magnetism and phase transitions.
  • Centered-triangular lattices introduce geometric frustration, leading to complex behaviors.

Purpose of the Study:

  • To analyze phase transitions in the Potts model on a centered-triangular lattice.
  • To explore the (K,J) phase diagram, focusing on the antiferromagnetic regime (K<0).

Main Methods:

  • Finite-size analysis using numerical transfer-matrix calculations.
  • Monte Carlo simulations to investigate the model's behavior.

Main Results:

  • Identified two phase transitions for all finite J when varying K.
  • Characterized critical properties, particularly in the geometrically frustrated antiferromagnetic case.
  • Observed algebraic phases with infinite-order transitions to the ferromagnetic phase in the J→±∞ limits.

Conclusions:

  • The centered-triangular Potts model exhibits rich phase transition behavior, especially under geometric frustration.
  • The interplay between couplings K and J dictates the model's critical phenomena.