Shell-model representations of the microscopic version of the Bohr–Mottelson collective model

The structure of the irreducible collective spaces of the group Sp(12, R), which many-particle nuclear states are classified according to the chain Sp(12, R) ⊃ U(6) ⊃ SO(6) ⊃ SUpn(3) ⨂ SO(2) ⊃ SO(3) of the proton–neutron symplectic model (PNSM), is considered in detail. This chain of the PNSM was recently shown to correspond to a microscopic shell-model version of the Bohr–Mottelson collective model. The construction of the relevant shell-model representations of the Sp(12, R) group along this chain is considered for three nuclei with varying collective properties and from different mass regions. It is shown that the SUpn(3) basis states of the Sp(12, R) representations belonging to the SO(6) irreps with seniority υ ≥ υ0, with υ0 denoting the maximal seniority SO(6) irrep contained in the Sp(12, R) bandhead, are always Pauli allowed, but organized in a different way into different SO(6) shells. This is in contrast to the case of filling the levels of the standard three-dimensional harmonic oscillator and using the plethysm operation. Although the SUpn(3) multiplets within υ < υ0 are not all Pauli forbidden, it is safe to discard them. The results obtained in the present work are important for the practical application of the microscopic version of the Bohr–Mottelson collective model.