Symmetry-Adaptation and Selection Rules for Effective Crystal Field Hamiltonians

  • J. A. Tuszyński

Abstract

The theory of nuclear and atomic shells owes its elegance to the group theoretic method of Racah1. He implemented a branching scheme leading to rotational subgroups of the unitary group U(4ℓ+2) of transformations among the spin-orbitals |nℓmsm >2. This subgroup reduction scheme induces analogous transformations among the many-particle states of the nucleus or the electronic shells. This results in a convenient labelling according to the irreducible representations of the rotational subgroups3 as |ℓNSLMSMLWUτ >. The corresponding Lie algebra of generators consists of double tensor operators4 whose (4ℓ+2)2 components (k1 = 0, 1; −k1 ≦ q1 ≦ k1; k2 = 0, 1, ..., 2ℓ −k2 ≦ q2 ≦ k2) span the full unitary group U(4ℓ+2). Their definition via commutation relations with angular momenta reflects the rotation properties of spherical harmonics and is supplemented by the normalization condition4: \( < s\ell \left\| {w^{k_1 k_2 } \left\| {s\ell > = \left[ {k_1 ,k_2 } \right]} \right.^{\frac{1} {2}} } \right. \). An extension of these operators into the many-body formalism requires defining them as sums of single-particle operators: . Consequently, the many-particle atomic or nuclear problem can be reduced to the evaluation of matrix elements of tensor operators between the eigenstates of compound angular momentum. In practice such calculations are facilitated by the use of Wigner-Eckart theorem and the recoupling properties of angular momenta. Furthermore, coefficients of fractional parentage5 which relate N-particle states to (N-1)-particle states render the method almost algorithmic.

Keywords

Angular Momentum Selection Rule Crystal Field Tensor Operator Radial Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    G. Racah, Phys. Rev. 62: 438 (1942).CrossRefGoogle Scholar
  2. 2.
    B. R. Judd, Symmetry properties of atomic structure, in: “Atomic Physics,” V. W. Hughes et al., eds., Plenum Press, New York (1968).Google Scholar
  3. 3.
    B. R. Judd, Adv. Atom. Molec. Phys. 7: 251 (1971).CrossRefGoogle Scholar
  4. 4.
    B. R. Judd, “Operator Techniques in Atomic Spectroscopy”, McGraw-Hill, New York (1967).Google Scholar
  5. 5.
    B. R. Judd, “Second Quantization and Atomic Spectroscopy”, The Johns Hopkins University Press, Baltimore (1967).Google Scholar
  6. 6.
    H. Watanabe, “Operator Methods in Ligand Field Theory”, Prentice Hall, Englewood Cliffs (1966).Google Scholar
  7. 7.
    L. Armstrong, Jr., “Theory of the Hyperfine Structure of Free Atoms”, Wiley-Interscience, New York (1971).Google Scholar
  8. 8.
    L. Armstrong, Jr., J. Math. Phys. 7: 1891 (1966).CrossRefGoogle Scholar
  9. 9.
    Lr. Armstrong, Jr., and S. Feneuille, Adv. Atom. Mclec. Phys. 10: 1 (1974).CrossRefGoogle Scholar
  10. 10.
    J. P. Desclaux, Comp. Phys. Commun. 9: 31 (1975).CrossRefGoogle Scholar
  11. 11.
    P. G. H. Sandars, and J. Beck, Proc. Roy. Soc. Lon. A289: 97 (1965).CrossRefGoogle Scholar
  12. 12.
    B. G. Wybourne, J. Chem. Phys. 43: 4506 (1965).CrossRefGoogle Scholar
  13. 13.
    R. Chatterjee, J. A. Tuszyński, and H. A. Buckmaster, Can. J. Phys. 61: 1613 (1983).CrossRefGoogle Scholar
  14. 14.
    J. A. Tuszyński, Physica A131: 289 (1985).CrossRefGoogle Scholar
  15. 15.
    J. L. Prather, “Atomic Energy Levels in Crystals”, U.S. National Bureau of Standards, Washington, D.C. (1961).Google Scholar
  16. 16.
    D. M. Brink and G. R. Satchler, “Angular Momentum”, Oxford University Press, London (1968).Google Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • J. A. Tuszyński
    • 1
  1. 1.Department of PhysicsMemorial University of NewfoundlandSt. John’sCanada

Personalised recommendations