Crystal-Field Effects not yet Fully Incorporated

  • C. A. Morrison
Part of the Lecture Notes in Chemistry book series (LNC, volume 47)


In section 12 we discussed the point-charge contribution to the multipolar field components Anm. It was early recognized by Hutchings and Ray (1963) that the multipolar components of the constituent ions contribute to the Anm at the site occupied by the unfilled shell ndN. For a point charge eZi located at (Math). from the origin ion site, we have the electric potential
$$\phi(\vec r)=\frac{{e{Z_i}}}{{|{{\vec R}_i}-\vec r|}}$$
. The potential energy of one of the ℓN electrons at r is
$$\begin{array}{*{20}{c}}{U(\vec r)=-ej(r)}\\{U(\vec r)=-{e^2}{Z_i}\sum\limits_{nm}{\frac{{{r^n}}}{{R_i^{n+1}}}{C_{nm}}(\hat r){C_{nm}}({{\hat R}_i})}}\\\end{array}$$
where we have expanded the denominator of equation (14.1) in the spherical tensors discussed in chapter 1. If we write equation (14.2) as
$$A_{nm}^{(0)}=-{e^2}\sum\limits_i{\frac{{{Z_i}{C_{nm}}({{\hat R}_i})}}{{R_i^{n+1}}}}$$
where the sum on i covers all the ions of charge eZi in the solid. This result we derived in section 11, expressed in slightly different form. It seems natural to extend equation (14.3) to the form
$$U(\vec r)=\sum\limits_{n,m,k}{A_{nm}^{(k)*}{r^n}{C_{nm}}}(\hat r)$$
and relate the A nm (k) to the various k-pole moments of ligands at \({\vec R_i}\).


Dipole Moment Multipole Moment Crystal Field Parameter Spherical Tensor Scheelite Structure 
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Annotated Bibliography and References

  1. Faucher, M., and P. Caro (1977), A Quickly Converging Method for Computing Electrostatic Crystal Field Parameters, J. Chem. Phys. 66, 1273.CrossRefGoogle Scholar
  2. Faucher, M., and D. Garcia (1983a), Crystal Field Effects in Rare-Earth-Doped Oxyhalides: Ab-initio Calculations Including Effects of Dipolar and Quadrupolar Moments, Solid State Chemistry, Proceedings of the Second European Conference (7–9 June 1982, Veldhoven, Netherlands), Elsevier, Amsterdam, Netherlands, p 547.Google Scholar
  3. Faucher, M., and D. Garcia (1983b), Crystal Field Effects on 4f Electrons: Theories and Reality, J. Less-Common Metals 93, 31.CrossRefGoogle Scholar
  4. Faucher, M., and D. Garcia (1982), Electrostatic Crystal-Field Contributions in Rare-Earth Compounds with Consistent Multipolar Effects: I.—Contribution to K-Even Parameters, Phys. Rev. B26, 5451.Google Scholar
  5. Garcia, D. (1983), Simulation ab-initio des parametres du champ des ligandes, thesis, L’Ecole Centrale des Arts et Manufactures.Google Scholar
  6. Garcia, D., and M. Faucher (1984), Crystal-Field Parameters in Rare-Earth Compounds: Extended Charge Contributions, Phys. Rev. A30, 1703.Google Scholar
  7. Garcia, D., M. Faucher, and O. L. Malta (1983), Electrostatic Crystal-Field Contribution in Rare-Earth Compounds with Consistent Multipolar Effects: II.—Contribution to K-Odd Parameters (Transition Probabilities), Phys. Rev. B27, 7386.Google Scholar
  8. Hutchings, M. T., and D. K. Ray (1963), Investigations into the Origin of Crystalline Electronic Field Effects on Rare-Earth Ions: I—Contributions from Neighbouring Induced Moments, Proc. Phys. Soc. 81, 663.CrossRefGoogle Scholar
  9. International Tables for X-Ray Crystallography (1952), Kynock Press, U.K.Google Scholar
  10. Jackson, J. D. (1975), Classical Electrodymanics, Wiley, New York, NY, chapters 3 and 4. Note that his multipole qℓm are related to our Qkq by \(q_{\ell m}^* = (\sqrt {(2\ell+ 1)/4\pi } ){Q_{\ell m}}\).Google Scholar
  11. Judd, B. R. (1977), Correlation Crystal Fields for Lanthanide Ions, Phys. Rev. Lett. 39, 242.CrossRefGoogle Scholar
  12. Morrison, C. A. (1980, January 15), Host Dependence of the Rare-Earth Ion Energy Separation 4FN — 4FN-1nℓ, J. Chem. Phys. 12, 1001.Google Scholar
  13. Morrison, C. A. (1976), Dipolar Contributions to the Crystal Fields in Ionic Solids, Solid State Commun. 18, 153.CrossRefGoogle Scholar
  14. Morrison, C. A., G. F. de Sa, and R. P. Leavitt (1982), Self-Induced Multipole Contribution to the Single-Electron Crystal Field, J. Chem. Phys. 76, 3899.CrossRefGoogle Scholar
  15. Rainville, E. D. (1960), Special Functions, Macmillan, New York, NY, p 1822.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • C. A. Morrison
    • 1
  1. 1.Harry Diamond LaboratoriesAdelphiUSA

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