Single-particle and collective motion in unbound deformed ${}^{39}\mathrm{Mg}$

Background: Deformed neutron-rich magnesium isotopes constitute a fascinating territory where the interplay between collective rotation and single-particle motion is strongly affected by the neutron continuum. The unbound $fp$-shell nucleus ${}^{39}\mathrm{Mg}$ is an ideal candidate to study this interplay.

Purpose: In this work, we predict the properties of low-lying resonant states of ${}^{39}\mathrm{Mg}$, using a suite of realistic theoretical approaches rooted in the open quantum system framework.

Method: To describe the spectrum and decay modes of ${}^{39}\mathrm{Mg}$ we use the conventional shell model, Gamow shell model, resonating group method, density matrix renormalization group method, and the nonadiabatic particle-plus-rotor model formulated in the Berggren basis.

Results: The unbound ground state of ${}^{39}\mathrm{Mg}$ is predicted to be either a $J^{\pi }={7/2}^{-}$ state or a ${3/2}^{-}$ state. A narrow $J^{\pi }={7/2}^{-}$ ground-state candidate exhibits a resonant structure reminiscent of that of its one-neutron halo neighbor ${}^{37}\mathrm{Mg}$, which is dominated by the $f_{7/2}$ partial wave at short distances and a $p_{3/2}$ component at large distances. A $J^{\pi }={3/2}^{-}$ ground-state candidate is favored by the large deformation of the system. It can be associated with the ${1/2}^{-}\left[321\right]$ Nilsson orbital dominated by the $\ell =1$ wave; hence its predicted width is large. The excited $J^{\pi }={1/2}^{-}$ and $5/2^{-}$ states are expected to be broad resonances, while the $J^{\pi }={9/2}^{-}$ and ${11/2}^{-}$ members of the ground-state rotational band are predicted to have very small neutron decay widths.

Conclusion: We demonstrate that the subtle interplay between deformation, shell structure, and continuum coupling can result in a variety of excitations in an unbound nucleus just outside the neutron drip line.

Figure 1

Fit of the GSM effective interaction for ${}^{34-40}\mathrm{Mg}$ carried out at the SM level. Experimental data [53] are denoted by stars. Because the masses of ${}^{38,39,40}\mathrm{Mg}$ have not yet been measured, for these nuclei we used the $S_n$ values (marked by filled stars) obtained from systematic trends, as quoted in Ref. [53]. The results of the optimization variants V1 and V2 are marked by blue and red lines, respectively.

Figure 2

Excited states of ${}^{39}\mathrm{Mg}$ predicted in different models used in this work (see text for more details).

Figure 3

Single-particle neutron Nilsson diagram for ${}^{39}\mathrm{Mg}$ from the relativistic mean-field approach with the complex-scaling method. The crossing between the deformed levels ${7/2}^{-}\left[303\right]$ and ${1/2}^{-}\left[321\right]$ originating from the $0f_{7/2}$ and $1p_{3/2}$ shells, respectively, results in a deformed subshell closure at $N=28$ and ${\beta }_2\approx 0.3$.

Figure 4

Convergence of the DMRG energies for (a) $J^{\pi }={7/2}^{-}$ and (b) ${3/2}^{-}$ states in ${}^{39}\mathrm{Mg}$ with respect to the number of included shells. The use of natural orbitals greatly improves the convergence and provides more consistent results when compared to the traditional DMRG calculation in WS orbitals. The inclusion of additional particle-hole excitations greatly improves the description of the $J^{\pi }={3/2}^{-}$ excited state, while it has little impact on the $J^{\pi }={7/2}^{-}$ g.s. The DMRG convergence criterion was assumed to be $\varepsilon ={10}^{-8}$. The RGM results are marked by a dotted line.

Figure 5

Partial-wave contributions of the $|0^{+}\rangle \otimes |f_{7/2}\rangle$ and $|2^{+}\rangle \otimes |p_{3/2}\rangle$ PRM channels to the decay width of the $J^{\pi }={7/2}^{-}$ resonance at $129\text{keV}$ in ${}^{39}\mathrm{Mg}$. The width corresponding to the imaginary part of the complex energy eigenvalue obtained by the direct diagonalization of the coupled-channel PRM equations is indicated by a dotted line.

Figure 6

One-body radial density of the valence neutron of the $J^{\pi }={7/2}^{-}$ resonance at $129\text{keV}$ in ${}^{39}\mathrm{Mg}$ in the PRM. The dashed and dotted lines represent the $f_{7/2}$ and $p_{3/2}$ contributions, respectively.