The proton-neutron symplectic model of nuclear collective motions

A proton-neutron symplectic model of collective motions, based on the non-compact symplectic group $Sp(12,R)$, is introduced by considering the symplectic geometry of the two-component many-particle nuclear system. The possible classical collective motions are determined by different dynamical groups that can be constructed from the symplectic generators. The relation of the $Sp(12,R)$ irreps with the shell-model classification of the basis states is considered by extending of the state space to the direct product space of $SU_{p}(3) \otimes SU_{n}(3)$ irreps, generalizing in this way the Elliott's $SU(3)$ model for the case of two-component system. The $Sp(12,R)$ model appears then as a natural multi-major-shell extension of the generalized proton-neutron $SU(3)$ scheme which takes into account the core collective excitations of monopole and quadrupole, as well as dipole type associated with the giant resonance vibrational degrees of freedom. Each $Sp(12,R)$ irreducible representation is determined by a symplectic bandhead or an intrinsic $U(6)$ space which can be fixed by the underlying proton-neutron shell-model structure, so the theory becomes completely compatible with the Pauli principle. It is shown that this intrinsic $U(6)$ structure is of vital importance for the appearance of the low-lying collective bands with both the positive and negative parity. The full range of low-lying collective states can then be described by the microscopically based intrinsic $U(6)$ structure, renormalized by coupling to the giant resonance vibrations.