Shape variables and the shell model

Abstract

The irreducible representation labelsλ andμ of the SU(3) shell model are related to the shape variablesβ andγ of the collective model by invoking a linear mapping between eigenvalues of invariant operators of the two theories. All but one parameter of the theory is fixed if the shell-model result is required to reproduce the collective-model geometry. And for one special value of the remaining free parameter there is a simple linear relationship between the eigenvalues, λ_α, of the quadrupole matrix of the collective model and the SU(3) representation labels:

$$\lambda _1 = ( - \lambda + \mu )/3, \lambda _2 = ( - \lambda + 2\mu + 3)/3, \lambda _3 = (2\lambda + \mu + 3)/3.$$

The correspondence between hamiltonians that describe rotations in each theory is also given. Results are shown for two cases,²⁴Mg and¹⁶⁸Er, to demonstrate that the simplest mapping yields excellent results for both energies and transition rates. For λ and/or μ large, the (β, γ)↔(λ,μ) correspondence introduced here reduces to the symplectic shell-model result.