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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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In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must...
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In mechanical engineering, a three-dimensional force system is a system of forces acting in three dimensions, with forces applied along the x, y, and z coordinate axes. The three-dimensional force system is an important concept in mechanical engineering, as it allows engineers to understand and analyze the behavior of objects and structures in three dimensions. By understanding the forces acting on a system, engineers can design more efficient and effective mechanical systems that can withstand...
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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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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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A three-dimensional force system refers to a scenario in which three forces act simultaneously in three different directions. This type of problem is commonly encountered in physics and engineering, where it is necessary to calculate the resultant force on the system, which can then be used to predict or analyze the behavior of the object or structure under consideration.
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Spin models in three dimensions: Adaptive lattice spacing.

Martin Hasenbusch1

  • 1Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 15, 2015
PubMed
Summary

This study introduces adaptive lattice spacing for studying critical phenomena near complex boundaries, successfully reproducing homogeneous system results for the 3D Ising model.

Area of Science:

  • Statistical Mechanics
  • Computational Physics

Background:

  • Studying critical phenomena near complex boundaries is challenging.
  • Adaptive lattice spacing is common in solving partial differential equations but difficult for Hamiltonians.

Purpose of the Study:

  • To implement adaptive lattice spacing for studying critical phenomena with nontrivial boundary shapes.
  • To focus on the universality class of the three-dimensional Ising model.

Main Methods:

  • Utilizing an improved Blume-Capel model on a simple cubic lattice.
  • Implementing sectors with varying lattice spacings (a, 2a, 4a,...).
  • Coupling lattice sectors and performing finite-size scaling studies.

Main Results:

  • Successfully coupled lattice sectors with different spacings.

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  • Identified and removed slowly decaying corrections by adjusting boundary couplings.
  • Determined magnetization profiles and thermodynamic Casimir forces.

    • Conclusions:

    • Adaptive lattice spacing can be consistently implemented for critical phenomena studies.
    • The method reproduces homogeneous system results, validating its applicability.
    • This approach is promising for investigating systems with complex boundary geometries.