Symmetry-Adaptation and Selection Rules for Effective Crystal Field Hamiltonians

  • J. A. Tuszyński


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.


Angular Momentum Selection Rule Crystal Field Tensor Operator Radial Integral 
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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

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