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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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Author Spotlight: Development of a Novel Finite Element Analysis Model for Improved Orthognathic Surgical Techniques
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L-based numerical linked cluster expansion for square lattice models.

Mahmoud Abdelshafy1, Marcos Rigol1

  • 1Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA.

Physical Review. E
|October 18, 2023
PubMed
Summary
This summary is machine-generated.

We developed a new L-shape cluster expansion for square-lattice models. This method offers improved convergence for spin-1/2 models, enabling lower-temperature calculations and studying lattice geometry connections.

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Area of Science:

  • Condensed Matter Physics
  • Computational Physics
  • Statistical Mechanics

Background:

  • Numerical linked cluster expansions are crucial for studying lattice models.
  • Traditional expansions often use larger clusters, impacting computational efficiency.
  • Investigating alternative cluster shapes can improve convergence and reduce costs.

Purpose of the Study:

  • To introduce and evaluate a novel numerical linked cluster expansion using an L-shape cluster as the building block.
  • To compare the performance of this L-shape expansion against traditional square-shaped clusters for spin-1/2 models.
  • To explore the applicability of the L-shape expansion for lattice models bridging square and triangular geometries.

Main Methods:

  • Development of a numerical linked cluster expansion with an L-shape cluster.
  • Application to various spin-1/2 lattice models.
  • Comparison of weak- and strong-embedding versions of the expansion.
  • Integration with resummation techniques (site- and bond-based expansions).

Main Results:

  • The L-shape cluster expansion demonstrates comparable or superior convergence of bare sums compared to larger square clusters.
  • The expansion facilitates calculations at significantly lower temperatures when combined with resummation techniques.
  • The strong-embedding version shows better convergence and lower computational cost than the weak-embedding version.
  • The L-shape cluster expansion naturally extends to study models connecting square and triangular lattice geometries.

Conclusions:

  • The L-shape cluster expansion is an efficient and effective method for studying spin-1/2 lattice models.
  • This approach offers advantages in convergence and computational cost, particularly the strong-embedding version.
  • The L-shape expansion provides a versatile tool for investigating lattice models with varying geometries.