The Effective Crystal Field Potential

The Effective Crystal Field Potential

1st Edition - June 22, 2000

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  • Editors: J. Mulak, Z. Gajek
  • Hardcover ISBN: 9780080436081
  • eBook ISBN: 9780080530710

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As it results from the very nature of things, the spherical symmetry of the surrounding of a site in a crystal lattice or an atom in a molecule can never occur. Therefore, the eigenfunctions and eigenvalues of any bound ion or atom have to differ from those of spherically symmetric respective free ions. In this way, the most simplified concept of the crystal field effect or ligand field effect in the case of individual molecules can be introduced.The conventional notion of the crystal field potential is narrowed to its non-spherical part only through ignoring the dominating spherical part which produces only a uniform energy shift of gravity centres of the free ion terms. It is well understood that the non-spherical part of the effective potential "seen" by open-shell electrons localized on a metal ion plays an essential role in most observed properties. Light adsorption, electron paramagnetic resonance, inelastic neutron scattering and basic characteristics derived from magnetic and thermal measurements, are only examples of a much wider class of experimental results dependent on it. The influence is discerned in all kinds of materials containing unpaired localized electrons: ionic crystals, semiconductors and metallic compounds including materials as intriguing as high-Tc superconductors, or heavy fermion systems. It is evident from the above that we deal with a widespread effect relative to all free ion terms except those which can stand the lowered symmetry, e.g. S-terms.Despite the universality of the phenomenon, the available handbooks on solid state physics pay only marginal attention to it, merely making mention of its occurrence. Present understanding of the origins of the crystal field potential differs essentially from the pioneering electrostatic picture postulated in the twenties. The considerable development of the theory that has been put forward since then can be traced in many regular articles scattered throughout the literature. The last two decades have left their impression as well but, to the authors' best knowledge, this period has not been closed with a more extended review. This has also motivated us to compile the main achievements in the field in the form of a book.


For experimentalists, particularly spectroscopists, and researchers involved in inelastic neutron scattering, magnetic measurements and the thermodynamics of solids. Also of interest to researchers in the areas of solid state physics and quantum chemistry.

Table of Contents

  • Chapter headings and selected sub-headings: Introduction. Parameterization of Crystal Field Hamiltonian. Operators and parameters of the crystal field Hamiltonian. Basic parameterizations. Symmetry transformations of the operators. The number of independent crystal field parameters. Standardization of the crystal field Hamiltonian. The Effective Crystal Field Potential. Chronological Development of Crystal Field Models. Ionic Complex or Quasi-Molecular Cluster. Generalized Product Function. Concept of the generalized product function. The density functions and the transition density functions. Model of the generalized product functions. Crystal field effect in the product function model. Point Charge Model (PCM). PCM potential and its parameters. Simple partial PCM potentials. Extension of PCM - higher point multipole contribution. One-Configurational Model with Neglecting the Non-Orthogonality. The Charge Penetration and Exchange Effects. Classical electrostatic potential produced by the ligand charge distribution. The charge penetration effect and the exchange interaction in the generalized product function model. The weight of the penetration and exchange effects in the crystal field potential. Calculation of the two-centre integrals. The Exclusion Model. One-Configurational Approach with Regard to Non-Orthogonality of the Wave Functions. Three types of the non-orthogonality. The contact-covalency - the main component of the crystal 84 field potential. The contact-shielding. The contact-polarization. Mechanisms of the contact-shielding and contact-polarization in terms of the exchange charge notion. Covalency Contribution, i.e. The Charge Transfer Effect. The one-electron excitations. Group product function for the excited state. The renormalization of the open-shell Hamiltonian due to the covalency effect. Basic approximations. Remarks on the covalency mechanism. Shielding and Antishielding Effect: Contributions from Closed Electron Shells. Phenomenological quantification of the screening effect. Microscopic model of the screening effect. General expressions for the screening factors. Electrostatic Crystal Field Contributions with Consistent Multipolar Effects. Polarization. Expansion of the electrostatic potential of point charge system into the multipole series. Extended formula for the crystal field parameters including all multipole moments of the surroundings. The self-consistent system of permanent and induced multipole moments in crystal lattice. The off-axial polarization terms in local coordinate systems.
    Crystal Field Effect in the Stevens Perturbation Approach. The perturbation scheme for degenerate systems employing projection operators. Specific Mechanisms of Metallic States Contributing to the Crystal Field Potential. Screening the Crystal Field in Metallic Materials. The Fourier form of the crystal lattice potential. Virtual Bound State Contribution to the Crystal Field Potential. The resonance scattering of conduction electrons field potential. Spin-polarization of the virtual bound state. The crystal field splitting of the virtual bound state. Corrections to the simple model of the virtual bound state mechanism. Hybridization or Covalent Mixing Between Localized States in Metallic Crystals. Hybridization contribution to the crystal field parameters. The Scale of the hybridization effect. Contribution to the crystal field potential from a split-off state from the conduction band in impurity systems. Density Functional Theory Approach. Electron density as a key variable. Local density approximation. Mapping DFT on effective Hamiltonian. Applications. Analysis of the Experimental Data. Interpretation of Crystal Field Parameters with Additive Models. Simplified crystal field models. Lattice Dynamics Contribution. Lattice dynamics and the crystal field effect. Extension of the Crystal Field Potential Beyond the One-Electron Model. Parameterization of the two-electron potential. Many-electron approach to the crystal field effect. Appendices. Author index. Keyword index.

Product details

  • No. of pages: 316
  • Language: English
  • Copyright: © Elsevier Science 2000
  • Published: June 22, 2000
  • Imprint: Elsevier Science
  • Hardcover ISBN: 9780080436081
  • eBook ISBN: 9780080530710

About the Editors

J. Mulak

Z. Gajek

Affiliations and Expertise

Institute of Low Temperature and Structure Research, Polish Academy of Sciences, ul. Okolna 2, 50-422 Wroclaw, Poland.

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