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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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Optimization of Crystal Growth for Neutron Macromolecular Crystallography
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Reductive renormalization of the phase-field crystal equation.

Y Oono1, Y Shiwa

  • 1Department of Physics, 1110 West Green Street, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-3080, USA. yoono@illinois.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 2, 2013
PubMed
Summary

This study unifies singular perturbation and reductive perturbation using renormalization-group theory. The reductive renormalization method simplifies complex systems, demonstrating consistency for differential equations and reducing a phase-field crystal equation.

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Area of Science:

  • Mathematical Physics
  • Computational Physics
  • Nonlinear Dynamics

Background:

  • Singular perturbation and reductive perturbation methods are crucial for simplifying complex systems.
  • Unifying these methods offers a more streamlined approach to analyzing dynamic equations.
  • Previous expositions of reductive renormalization have been cryptically presented, hindering practical application.

Purpose of the Study:

  • To demonstrate the consistency of the reductive renormalization-group procedure for partial differential equations.
  • To apply the reductive renormalization method to a phase-field crystal equation, illustrating its practical utility.
  • To provide a clearer understanding and accessible exposition of the reductive renormalization technique.

Main Methods:

  • Renormalization-group (RG) theory to unify perturbation methods.
  • Reductive renormalization procedure for extracting global behavior.
  • Application to partial differential equations, including time-evolution semigroup types.
  • Illustrative reduction of a phase-field crystal equation.

Main Results:

  • Explicit demonstration of the consistency of the reductive renormalization-group procedure.
  • Successful application of the method to simplify a phase-field crystal equation.
  • The reductive renormalization method is shown to be simpler than standard scaling expansions.

Conclusions:

  • The reductive renormalization-group procedure is a consistent and effective method for system reduction.
  • This unified approach offers a streamlined alternative to traditional perturbation methods.
  • Future work may explore the diffeomorphic relationship between RG results and original equations for structurally stable systems.