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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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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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Crystal Field Theory
To explain the observed behavior of transition metal complexes (such as colors), a model involving electrostatic interactions between the electrons from the ligands and the electrons in the unhybridized d orbitals of the central metal atom has been developed. This electrostatic model is crystal field theory (CFT). It helps to understand, interpret, and predict the colors, magnetic behavior, and some structures of coordination compounds of transition metals.
CFT focuses on...
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In the macroscopic world, objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle; it will continue traveling in a straight line unless it collides with another ball, or it is acted on by some other force, such as friction. The ball has a well-defined position and velocity or well-defined momentum, p = mv, which is defined by mass m and velocity v at any given moment. This is the typical...
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Shortly after de Broglie published his ideas that the electron in a hydrogen atom could be better thought of as being a circular standing wave instead of a particle moving in quantized circular orbits, Erwin Schrödinger extended de Broglie’s work by deriving what is now known as the Schrödinger equation. When Schrödinger applied his equation to hydrogen-like atoms, he was able to reproduce Bohr’s expression for the energy and, thus, the Rydberg formula governing hydrogen spectra.
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Microscopic lattice model for quartic semi-Dirac fermions in two dimensions.

Mohamed Marwan Elsayed1, Valeri N Kotov1

  • 1Department of Physics, University of Vermont, Burlington, VT 05405, United States of America.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|July 16, 2025
PubMed
Summary

We introduce a lattice model for exotic quartic semi-Dirac fermions. Short-range interactions are crucial for stabilizing this phase, preventing transitions to Dirac cones or quadratic semi-Dirac phases.

Keywords:
electron–electron interaction effectssemi-Dirac fermionstunable lattice modelstwo dimensional systems

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Area of Science:

  • Condensed Matter Physics
  • Solid State Physics
  • Quantum Materials

Background:

  • Semi-Dirac fermions possess unique linear and quadratic dispersion relations.
  • Realizing exotic phases like quartic semi-Dirac fermions requires specific lattice and interaction engineering.

Purpose of the Study:

  • To propose and investigate a lattice model for realizing quartic semi-Dirac fermions.
  • To identify the conditions necessary for stabilizing this exotic phase.

Main Methods:

  • Utilizing a tight-binding model with up to fourth nearest-neighbor hopping.
  • Incorporating anisotropic hopping parameters to control lattice properties.
  • Introducing short-range electron-electron interactions.

Main Results:

  • The proposed model successfully realizes quartic semi-Dirac fermions.
  • Short-range interactions are demonstrated to be essential for stabilizing the quartic semi-Dirac phase.
  • Without interactions, or with long-range correlations, the system transitions to anisotropic Dirac cones or quadratic semi-Dirac phases.

Conclusions:

  • A stable lattice model for quartic semi-Dirac fermions is presented.
  • Electron-electron interactions play a critical role in stabilizing exotic electronic phases.
  • The findings provide a pathway for exploring novel quantum phenomena in engineered materials.