The states |v,α,L,M⟩ and SO(5) spherical harmonics Υ_v,α,L,M are labelled by SO(5) seniority v, SO(3) angular momentum L and magnetic component M, with α a "missing label" that distinguishes angular momentum L irreps in the SO(3) reduction of seniority v irreps of SO(5).

The seniority v representation of SO(5) is the "one-rowed" irrep of SO(5) having Dynkin label (v,0). It has dimension (v+1)(v+2)(2v+3)/6. On restricting this irrep to the particular subgroup SO(3) of SO(5) defined by the Bohr model [Bo1952] (see [RW2010] for a modern overview) of the atomic nucleus, the SO(3) irrep of angular momentum L occurs with a multiplicity d_v,L given by d_v,L= (⌊(v-b)/3⌋+1)θ_v-b -⌊(v-L+2)/3⌋θ_v-L+2, where b=L/2 for L even, and b=(L+3)/2 for L odd, and we define θ_k=1 for k≥0, and θ_k=0 for k<0. The missing label then takes the range 1≤α≤d_v,L. Lists of valid labels (v,L;α) may be found here. Expressions for the SO(5) spherical harmonics Υ_v,α,L,M may be found here.

Each SO(5) Clebsch-Gordan coefficient (v₁,α₁,L₁,M₁; v₂,α₂,L₂,M₂ | v₃,α₃,L₃,M₃) may be expressed

(v1,α1,L1,M1; v2,α2,L2,M2 | v3,α3,L3,M3) = (v1,α1,L1; v2,α2,L2 || v3,α3,L3) (L1,M1; L2,M2 | L3,M3),

via the Wigner-Eckart theorem. Here, (L₁,m₁; L₂,m₂ | L₃,m₃) is a usual SO(3) Clebsch-Gordan coefficient, and (v₁,α₁,L₁; v₂,α₂,L₂ || v₃,α₃,L₃) is an SO(5)⊃SO(3) reduced Clebsch-Gordan coefficient. Their use in the Algebraic Collective Model of atomic nuclei is expounded and illustrated in [RWC2009] and [RW2010].

Data files of SO(5)⊃SO(3) Clebsch-Gordan coefficients have been compiled using the algorithm developed in [RTR2004], and refined in [CRW2009]. Note that the ordering employed in the generation of these components is that in the latter of these papers (this choice arises from the multiplicity in the spaces of constant v and L). These files are utilised in our Maple implementation of the Algebraic Collective Model, which is described in [WR2016].

References

[Bo1952] "The coupling of nuclear surface oscillations to the motion of individual nucleons", by A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26:14 (1952).

[RTR2004] "Spherical harmonics and basic coupling coefficients for the group SO(5) in an SO(3) basis", by D.J. Rowe, P.S. Turner and J. Repka, J. Math. Phys. 45 (2004) 2761-2784.

[CRW2009] "Construction of SO(5) ⊃ SO(3) spherical harmonics and Clebsch-Gordan coefficients", by M.A. Caprio, D.J. Rowe and T.A. Welsh, Comp. Phys. Comm. 180 (2009) 1150-1163. (arXiv)

[RWC2009] "The Bohr Model as an algebraic collective model", by D.J. Rowe, T.A. Welsh and M.A. Caprio, Phys. Rev. C79 (2009) 054304.

[RW2010] "Fundamentals of Nuclear Models: I Foundational Models", by D.J. Rowe and J.L Wood, (World Scientific, Singapore), 2010.

[WR2016] "A computer code for calculations in the algebraic collective model of the atomic nucleus", by T.A. Welsh and D.J. Rowe, Comp. Phys. Comm. 200 (2016) 220-253. (arXiv)

Last updated: 19/3/2016.