Collectivization of anti-analog strength above charged particle thresholds

Ten years ago, highly excited states were found in ${}^9\mathrm{Li}$ and ${}^{10}\mathrm{Be}$ a few hundred kilovolts above the proton decay threshold. These physical states are too low in energy to be the isospin-stretched configuration of the decay channel (the isobaric analog or $T_{>}$). However, these states can be understood by a continuum cognizant shell model as strongly mixed states of lower isospin ($T_{<}$), where the mixing is largely mediated by the open neutron channels but ushered in energy to be just above the proton threshold.

Figure 1

Invariant-mass reconstructions of (a) ${}^9\mathrm{Li}^{*}$ and (b) ${}^{10}\mathrm{Be}^{*}$ from the indicated decay channels. The vertical lines are the proton-decay thresholds (13.944 and 19.636 MeV) and the dotted (dashed) curves indicate the fitted states (backgrounds).

Figure 2

Continuum-shell-model level diagrams for relevant (a) ${}^9\mathrm{Li}^{*}$ and (b)${}^{10}\mathrm{Be}^{*}$ resonances. All shown levels are coherently coupled in the SMEC calculations. The black (red) lines indicate $T_{<}$ ($T_{>}$) levels, lines with end points are experimental levels, and those with arrows indicate the lowest-lying levels with a significant proton-decay branch. The ${}^9\mathrm{Li}$ levels of $J^{\pi }=1/2^{+}$, ${}^{10}\mathrm{Be}$ levels with $J^{\pi }=2^{-}$ (solid lines) and $1^{-}$ (dashed lines), the ground-state proton decay channel, and the relevant $n$-decay channels are all indicated. The wave-function compositions are provided in the supplementary tables that are organized by the indices n and k [36].

Figure 3

The continuum-coupling correlation energy as a function of proton energy $E_p$ for (a) $J^{\pi }=1/2^{+}$, $T=3/2$ SM states with an excitation energy $E^{*}<15$ MeV coupled in $s$ wave to the (lowest) one-proton decay channel and (b) with the additional coupling to all one-neutron decay channels, and (c) ${}^{10}\mathrm{Be}^{*}$ for the $J_n^{\pi }=2_{12}^{-}$, $T=1$ state considering all $J^{\pi }=2^{-}$, $T=1$ SM states with an excitation energy $E^{*}<22$ MeV $s$ wave coupled to the (lowest) one-proton decay channel and (d) with the addition of coupling to nine neutron-decay channels.