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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.
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Updated: Dec 10, 2025

Neutron Crystallography Data Collection and Processing for Modelling Hydrogen Atoms in Protein Structures
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Self-consistent modeling of anisotropic interfaces and missing orientations: Derivation from phase-field crystal.

N Ofori-Opoku1,2, J A Warren2, P W Voorhees1,3,4

  • 1Center for Hierarchical Materials Design, Northwestern University, Evanston IL 60208.

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|September 1, 2020
PubMed
Summary

This study uses the phase-field crystal (PFC) model to quantitatively describe highly anisotropic interfaces. The model accurately predicts Wulff shapes, missing orientations, and facet formation in materials.

Keywords:
anisotropycoarse-grainingphase-field crystalsolidificationsurface energy

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Area of Science:

  • Materials Science
  • Computational Physics
  • Crystallography

Background:

  • Highly anisotropic interfaces are crucial for material microstructure development.
  • Quantitative modeling of these interfaces is essential for materials design.

Purpose of the Study:

  • To assess the phase-field crystal (PFC) formalism's capability in quantitatively describing highly anisotropic interfaces.
  • To coarse grain the PFC model into its complex amplitude and phase-field limits.

Main Methods:

  • Utilized the diffusive atomistic phase-field crystal (PFC) formalism.
  • Coarse-grained the PFC model to obtain complex amplitude and phase-field formulations.
  • Performed one-dimensional calculations to determine surface energy and Wulff shape properties.
  • Extended the model to two dimensions to study crystal growth.

Main Results:

  • The phase-field limit of the PFC model accurately describes anisotropic surface properties dependent on crystal orientation.
  • The model predicts Wulff shapes with missing orientations and facet formation.
  • Demonstrated the model's capability to study crystal growth with varying anisotropy in 2D.

Conclusions:

  • The coarse-grained PFC model provides a self-consistent description of highly anisotropic surface properties.
  • The phase-field limit naturally incorporates regularization and describes missing orientations in equilibrium crystal shapes.
  • The developed model is suitable for studying anisotropic crystal growth.