Symmetry-Adaptation and Selection Rules for Effective Crystal Field Hamiltonians

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 |nℓm_sm_ℓ >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 |ℓ^NSLM_SM_LWUτ >. The corresponding Lie algebra of generators consists of double tensor operators⁴ EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4DamaaCa % aaleqabaGaae4AamaaBaaameaacaqGXaaabeaaliaabUgadaWgaaad % baGaaeOmaaqabaaaaaaa!3AC5!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${{\text{w}}^{{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}}$$ whose (4ℓ+2)² components EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4DamaaDa % aaleaacaqGXbWaaSbaaWqaaiaabgdaaeqaaSGaaeyCamaaBaaameaa % caqGYaaabeaaaSqaaiaabUgadaWgaaadbaGaaeymaaqabaWccaqGRb % WaaSbaaWqaaiaabkdaaeqaaaaaaaa!3E86!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\text{w}}_{{{\text{q}}_{\text{1}}}{{\text{q}}_{\text{2}}}}^{{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}$$ (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: EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4DamaaCa % aaleqabaGaae4AamaaBaaameaacaqGXaaabeaaliaabUgadaWgaaad % baGaaeOmaaqabaaaaOGaaGPaVlabggMi6oaaqahabaGaae4DamaaDa % aaleaacaqGPbaabaGaae4AamaaBaaameaacaqGXaaabeaaliaabUga % daWgaaadbaGaaeOmaaqabaaaaaWcbaGaaeyAaiabg2da9iaaigdaae % aacaqGobaaniabggHiLdaaaa!49A0!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${{\text{w}}^{{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}}\, \equiv \sum\limits_{{\text{i}} = 1}^{\text{N}} {{\text{w}}_{\text{i}}^{{{\text{k}}_{\text{1}}}{{\text{k}}_{\text{2}}}}} $$. 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.