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A unique decay process: beta delayed emission of a proton and a neutron by the $^{11}$Li halo nucleus

The neutron-rich $^{11}$Li halo nucleus is unique among nuclei with known separation energies by its ability to emit a proton and a neutron in a $β$ decay process. The branching ratio towards this rare decay mode is evaluated within a three-body model for the initial bound state and with Coulomb three-body final scattering states. The branching ratio should be comprised between two extreme cases, i.e.\ a lower bound $6 \times 10^{-12}$ obtained with a pure Coulomb wave and an upper bound $5 \times 10^{-10}$ obtained with a plane wave. A simple model with modified Coulomb waves provides plausible values between between $0.8 \times 10^{-10}$ and $2.2 \times 10^{-10}$ with most probable total energies of the proton and neutron between 0.15 and 0.3 MeV.