Uncovering the nature of low-lying dipole states with QRPA calculations: is Z=42 the answer?

Abstract: The pygmy dipole resonance (PDR), marked by enhanced electric dipole strength near particle emission energies, offers a unique perspective on the collective dynamics of nuclear structure. Its precise nature, particularly its degree of collectivity, remains a topic of debate. In this study, we investigate low-energy dipole excitations in spherical Mo isotopes ($^{82}$Mo to $^{98}$Mo) using a fully consistent Hartree-Fock-Bogoliubov (HFB) and quasiparticle random phase approximation (QRPA) framework. We observe that an enhancement in dipole strength near particle emission energies is closely correlated with the development of either neutron or proton skins. To further understand the nature of this enhancement, we examine the behavior of proton and neutron transition densities. Our analysis shows that these (low-lying dipole) states exhibit distinct characteristics involving in-phase oscillations within the nucleus and neutron- or proton-dominated oscillations at the surface, while the primary contributor to this enhancement displays an intricate underlying structure. We also investigate the collectivity of these excitations by analyzing two-quasiparticle fragmentations and relative energy shifts. Our findings reveal that skin oscillation states exhibit moderate collectivity, as indicated by substantial configuration mixing, but limited coherence, whereas the GDR states exhibit strong coherence and large energy shifts characteristic of fully developed collective motion. This study paves the way for future investigations into the collective nature of low-energy dipole states in the enhancement region, particularly in deformed nuclei, where nuclear shape effects may play a crucial role in their excitation dynamics.