Observation of excited states in $^{20}$Mg sheds light on nuclear forces and shell evolution

TL;DR

This study reports the first observation of excited states in the proton-rich nucleus $^{20}$Mg, providing insights into nuclear forces, shell evolution, and potential weakening of the N=8 shell closure at the proton drip-line.

Contribution

It presents the first experimental detection of resonances and bound states in $^{20}$Mg, challenging current nuclear models and suggesting the need for improved theoretical approaches.

Findings

Observation of proton unbound resonances and bound states in $^{20}$Mg.

Experimental data challenge existing nuclear structure models.

Indication of weakening of the N=8 shell closure at the proton drip-line.

Abstract

The exotic Borromean nucleus $^{20}$Mg with $N$ = 8, located at the proton drip-line provides a unique testing ground for nuclear forces and the evolution of shell structure in the neutron-deficient region. We report on the first observation of proton unbound resonances together with bound states in $^{20}$Mg from the $^{20}$Mg($d$,$d'$) reaction performed at TRIUMF. Phenomenological shell-model calculations offer a reasonable description. However, our experimental results present a challenge for current first-principles nuclear structure approaches and point to the need for improved chiral forces and {\it ab initio} calculations. Furthermore, the differential cross section of the first excited state is compared with distorted-wave Born approximation calculations to deduce a neutron quadrupole deformation parameter of $\beta_n$=0.46$\pm$0.21. This provides the first indication of a possible weakening of the $N$ = 8 shell closure at the proton drip-line.

Full text

The evolution of shell structure over the nuclear landscape is a manifestation of strong interactions in the complex nuclear many-body system. Properties of nuclei at the neutron and proton drip-lines provide new arenas to investigate the effects of large proton-neutron asymmetry and understand the persistence of mirror symmetry. Shell structure evolution in neutron-rich and proton-rich nuclei JA05 ; JA09 ; OZ00 ; HO08 ; KA09 ; PI01 ; NA00 ; MO95 ; BA07 ; WIE13 ; ST13 ; GAR16 are leading to new insights into nuclear forces, including the role of three-body forces OT10 ; HEB15 . The region around the $N$ = 8 shell closure draws particular interest, since this shell gap disappears at the neutron drip-line and leads to the formation of a two-neutron halo in the Borromean nucleus 11Li. The small two-neutron separation energy ($S_{2n}$ = 360 keV) of 11Li results in the excited states of 11Li being unbound. Less is known about the structure of nuclei at the proton drip-line. The $N$ = 8 isotone at the proton drip-line, 20Mg, is also a Borromean system whose two-proton separation energy is $S_{2p}$ = 2.337(27) MeV AME12 . There is no experimental information on resonances above the proton threshold in 20Mg. In this work we present the first observation of a resonance in 20Mg through deuteron inelastic scattering. This measurement provides new insight into shell evolution as well as tests of ab initio predictions. The resonance(s) in 20Mg could also contribute to a potential breakout path from the hot CNO cycle via two-proton capture on the waiting point nucleus 18Ne in Type-I x-ray bursts GO95 .

Microscopic cluster model DE98 predictions for the 2+ state agree with the experiment GA07 ; IW08 . A 4+ state is predicted at 3 MeV, this state is predicted around 3.8 MeV, using a beyond the mean field approach RO16 . Predictions based on an angular momentum projected generator coordinate framework RO02 , find the 2${}_{1}^{+}$ and the 4${}_{1}^{+}$ states to be higher in energy around, 3.5 MeV and 7.8 MeV, respectively. Ab initio calculations in a many-body perturbation theory (MBPT) framework including the three-nucleon (3N) force HO13 predict the first excited state in fair agreement with Refs.DE98 ; RO16 . The predictions widely vary however, for states above the proton threshold with the $2_{2}^{+}$ and 4${}_{1}^{+}$ states being nearly degenerate placed around 4.2 MeV. Experimental information on these states above the proton threshold, is therefore necessary for testing the nuclear structure models and nuclear forces.

In addition, mass measurements of 20,21Mg GA14 cannot be reconciled with the known isobaric mass multiplet equation for 20Mg from isospin symmetric shell-model Hamiltonians BA06 or with that including isospin non-conserving interactions OR89 . The ab initio predictions based on MBPT HO13 show a stronger isospin dependence of the IMME than is experimentally observed GA14 . The energy of the lowest $T$ = 2, $J^{\pi}$ = $0^{+}$ state in 20Na measured from superallowed $\beta^{+}$ decay of 20Mg however successfully describes the $T$ = 2, mass 20 multiplet by a quadratic IMME GL15 . Excited states of the mirror nuclei 20Mg and 20O can provide further insight into the isospin dependence of the nuclear interaction and the nature of the $N$ = 8 shell closure at the proton drip-line.

The first excited state of 20Mg is a bound state that was observed first through gamma ray detection from the 9Be(22Mg,20Mg+$\gamma$)X reaction GA07 exhibiting a peak at 1.598(10) MeV. A gamma transition at 1.61(6) MeV was also observed in the Coulomb excitation of 20Mg with a Pb target at 58.4$A$ MeV IW08 . Assuming this as the 2+ state, a B(E2; 0${}^{+}\rightarrow$ 2+) value of 177(32) e2fm4 was deduced IW08 , which is in good agreement with predictions in Refs.RO02 as well as the cluster model predictions DE98 . In order to obtain insight on the deformation of neutrons in this $N$ = 8 isotone, inelastic scattering with a hadronic probe is needed together with information derived from Coulomb excitation.

In this article, we report the first study of deuteron inelastic scattering on 20Mg, populating new resonances together with the ground and first excited states in this proton drip-line nucleus. The experiment was carried out using the IRIS reaction spectroscopy facility KA13 at TRIUMF. A schematic view of the detector layout is shown in Fig.1. The radioactive beam of 20Mg was produced via fragmentation of a SiC target with 480 MeV protons. The beam was re-accelerated using the ISAC-II superconducting LINAC Bob to 8.5$A$ MeV and then passed through an ionization chamber, filled with isobutane gas at 19.5 Torr at room temperature. The energy loss of the beam, measured in this detector, provided an event by event identification of the 20Mg incident beam and its contaminant 20Na throughout the experiment. Following this, the beam interacted with a thin windowless solid deuterium 2H ($D_{2}$) reaction target built on a 4.5 $\mu$m thick Ag foil backing facing upstream of the $D_{2}$ layer. The target cell with the foil was cooled to 4 K, before forming solid $D_{2}$. The energy of the elastic scattered beam on the Ag foil was measured with and without $D_{2}$, providing a continuous measurement of the target thickness during the experiment. These scattered beam particles were detected using a double sided silicon strip detector placed 33 cm downstream of the target, covering laboratory angles 1.9*∘* to 6.1*∘. The average target thickness was 65 $\mu$m, and the value at each instant of time (each data collection run) was used for determining the scattering cross sections. The deuterons scattered out from reactions were detected using annular arrays of 100 $\mu$m thick single sided silicon strip detectors followed by a layer of 12 mm thick CsI(Tl) detectors. This detector combination served as an energy-loss ($\Delta$E) and total energy (E) telescope for identifying the $p,d,t$ and He recoils after the target. The CsI(Tl) detectors were calibrated using the elastic and inelastic scattering from a beam of 20Ne. The detector telescope covered scattering angles of $\theta_{lab}$ = 30.1∘* to 56.2*∘*.

The excitation spectrum of 20Mg, shown in Fig.2a, was reconstructed using the missing mass technique from the energy and scattering angle of the deuterons, measured by the silicon-CsI(Tl) telescope. The deuteron scattering can detect proton unbound resonances with no background from decay protons, unlike proton inelastic scattering. The ground state peak from 20Mg($d$,$d$) elastic scattering is clearly visible together with two prominent peaks, one below and one above the proton threshold. The background from the Ag foil was measured by collecting data without $D_{2}$ and is shown with the green dashed-dotted histogram normalized by the incident beam intensity. This background contribution was $\approx$ 10% of the total spectrum. The non-resonant background from 20Mg+$d\rightarrow^{18}$Ne+$p$+$p$+$d$ estimated from Monte Carlo simulation, including the detection conditions, is shown by the blue short dashed histogram. The total background from these two contributions is depicted by the red dashed histogram and is normalized to the data in the excitation energy region greater than 6.5 MeV.

The background subtracted excitation spectrum (Fig.2b), was fitted by a sum of two Gaussians, for the two bound state peaks, and two Breit-Wigner functions folded by Gaussian profiles accounting for the detection resolution, for the unbound resonance peaks. The excitation energy resolution ($\sigma$) for the ground state peak was 0.71 MeV for $\theta_{lab}$= 35*∘* and 0.45 MeV for $\theta_{lab}$= 50*∘*, which is in fair agreement with simulations. The resolution improves for the excited states to 0.48 MeV for the newly observed second excited state as determined based on simulations. The excitation energy of the first excited state peak is found to be 1.65${}^{+0.02}_{-0.10}$ MeV which is in good agreement with the observations in Refs.IW08 ; GA07 . Above the proton threshold, the most prominent peak is observed at an excitation energy of 3.70${}^{+0.02}_{-0.20}$ MeV with an apparent resonance width of 0.47$\pm$0.06 MeV after unfolding the resolution. The excitation energy uncertainties include observable peak shifts for possible systematic effects besides peak fitting. A small structure is also observed at a higher excitation energy of 5.37$\pm$0.02 MeV where the uncertainty quoted is from fitting only.

The energies of the first two excited states states observed in 20Mg are in close agreement with those of the mirror nucleus 20O LA73 . Shell model calculations with the phenomenological USDB BA06 interaction without any isospin dependence are in fair agreement with the data (see Fig. 3), though the first excited state is predicted slightly higher in energy and the second excited state observed seems closer to the predicted 4+ state. Calculations with an improved USDB interaction, including the Coulomb corrections and isospin dependence of Ref. HET89 , provide excellent agreement with the first excited (2+) state while the observed second excited state is midway between the predicted 4${}^{+}_{1}$ and 2${}_{2}^{+}$ states. Given the small predicted energy difference of these states, it may be possible that the new resonance peak observed in the experiment is an overlap of these two states. Furthermore, valence-space calculations within the MBPT framework, based on NN+3N forces HO13 (within an extended proton valence space that included the $sd$ shell plus $f_{7/2}$ and $p_{3/2}$ orbitals), also predict the first excited $2^{+}$ energy in good agreement with experiment, but higher-lying states are several hundred keV above the observed resonances. This could be expected since these calculations neglect effects of continuum coupling, which typically lower states by a few hundred keV.

New ab initio calculations were therefore performed. In particular, we use the valence-space in medium similarity renormalization group (VS-IMSRG) approach TSU12 ; BOG14 ; STR16 , and the coupled-cluster effective interaction (CCEI) method JA14 ; JA16 . The VS-IMSRG approach constructs an approximate unitary transformation MOR15 ; HER16 to first decouple the 16O core, as well as an $sd$ valence-space Hamiltonian, diagonalized using the NuShellX@MSU shell-model code BRO14 . We further capture the bulk effects of 3N forces between valence nucleons with the ensemble normal-ordering procedure of Ref. STR17 , thereby producing a unique valence-space Hamiltonian for each nucleus to be studied. This allows us to test nuclear forces in essentially any fully open-shell system accessible to the nuclear shell model with a level of accuracy comparable to large-space ab initio methods. In the CCEI approach, similar to the VS-IMSRG method, one calculates the valence-space effective interaction starting from a chiral NN+3N interaction and applies the obtained zero-plus-one-plus-two-body interaction in the standard shell model diagonalization. The CCEI approach JA16 utilizes coupled-cluster method to perform ab initio calculations for $A_{c}$, $A_{c}+1$ and $A_{c}+2$ system with $A_{c}$ the number of nucleons in the core (here $A_{c}=16$). The Okubo-Lee-Suzuki similarity transformation OK54 ; SU82 is then used to obtain an effective $sd$-shell interaction.

We take several sets of initial NN+3N forces in this work. The first, EM(1.8/2.0) HEB11 ; SIM16 , begins from the chiral NN N3LO force EN03 with a nonlocal 3N force fit in $A$ = 3 and 4-body systems, but reproduces ground- and excited-state energies to the tin region and beyond SIM17 ; MOR18 . The second, NN+3N(400) begins from the same NN force, but with local 3N forces NA07 ; RO12 , yields accurate binding energies and spectra in and around the oxygen isotopes BOG14 ; JA14 ; HE13 ; CI13 . Next, we utilized a newly developed chiral potential at N4LO PhysRevC.96.024004 combined with an N2LO 3N interaction with parameters fitted to the 3H binding energy and half-life (NN-N4LO+3N(lnl)). Finally N2LOsat has been fit to medium-mass data and reproduces ground-state energies and radii to the nickel region EKS15 .

VS-IMSRG calculations based on the EM(1.8/2.0) interaction accurately predicts the energy of the first $2^{+}$ state, but states above the proton threshold are lower than the experimental observations as well as the shell model predictions. The N2LOsat interaction has the worst agreement predicting the first excited state at only 600 keV. The other two interactions, NN+3N400 and NN N4LO+3N(lnl), provide similar results as EM(1.8/2.0), but with slightly lower 2${}^{+}_{1}$ energies and a greater spread in higher-lying states.

The MBPT results use older generation NN+3N forces and an extended valence space. We therefore compare the VS-IMSRG and CCEI frameworks where very similar results are obtained with the same forces, with the VS-IMSRG spectrum being modestly more spread. This is likely due to the inclusion of repulsive 3N forces between the four valence protons via the ensemble normal ordering procedure in the VS-IMSRG approach. There is a greater variance in the predicted energies for different nuclear forces.

We note here that, binding energy from the CCEI calculations with NN+3N(400) for 20Mg agree with the experiment. To further check the VS-IMSRG and CCEI calculations, we also performed no-core shell model (NCSM) BA13 calculations with the same interactions. For technical reasons, the largest basis space we could reach was $N_{\rm max}=6$ (utilizing the importance-truncation approach RO07 ). Such a space is too small to reach convergence and obtain a reliable estimate of excitation energies. However, our NCSM results within their uncertainties were consistent with the valence-space methods for binding energy as well as for the $2^{+}_{1}$ state excitation energy.

The observed unbound state of 20Mg therefore, points to the need for refinement of the nuclear forces although the IMSRG framework with the EM interaction (3N-full) in Ref.STR16 was shown to explain the spectra for Ne and F isotopes. While, the role of continuum effects needs to be assessed it should be noted that coupling to the continuum generally lowers the excitation energy as shown in Ref.HA16 . Furthermore, the shell model predictions without any coupling to the continuum agree well with the data.

The differential cross sections of the ground state, first and second excited states were derived from the area under the background subtracted peaks for different angular bins (Fig.4). The Woods-Saxon shape optical potential parameters for the 20Mg+$d$ interaction were determined from the best fit of DWBA calculations to the elastic scattering angular distribution data (Fig.4a) using sFRESCO TH . These parameters were then used for calculating the angular distributions for the excited states. The angular distribution of the first excited state (Fig.4b) is consistent with a multipolarity of excitation $L$ = 2, thereby determining for the first time its spin of 2+. The normalization of the calculation to the data provides the deformation length $\delta$ to be 1.33$\pm$0.23 fm, where $\delta={\frac{Z}{A}}\delta_{p}+{\frac{N}{A}}\delta_{n}$ with $\delta_{p}$ and $\delta_{n}$ being the proton and neutron deformation lengths, respectively. The deformation parameter is given by $\beta$ = $\delta$/$R$ with $R$ being the radius. $\delta_{p}$ was determined to be 1.32$\pm$0.12 fm from the quadrupole proton deformation parameter 0.44(4) derived from Coulomb excitation IW08 and a proton radius value of 3 fm that is consistent with theoretical predictions Gai14 ; WA07 ; LA01 . Therefore, a quadrupole deformation parameter for neutrons, $\beta_{n}$ = 0.46$\pm$0.21 is found. This large non-zero value within one standard deviation uncertainty, depicts neutron deformation and therefore a first signature of possible weakening of the $N$ = 8 shell closure. The $\beta_{n}$ for 20Mg is in agreement with $\beta$ of the mirror nucleus 20O found from proton scattering JE99 . Using the prescription of JE99 $\beta_{n}$ = 0.46$\pm$0.14 for 20Mg. The proton deformation parameter of 20O obtained from B(E2) measurements AD16 is also in agreement.

The angular distribution for the second excited state (Fig.4c), is not explained by either $L$ = 2 ($\chi^{2}_{red}\approx$ 2.7) or $L$ = 4 ($\chi^{2}_{red}\approx$ 4.1) excitation. There may be a possibility that the resonance peak has a mixture of $L$ = 2 and 4. The observed resonance energy lies between the predicted 2${}_{2}^{+}$ and 4+ states with the isospin dependent USDB interaction reflecting the importance of isospin dependence.

In summary, the first observation of proton unbound excited states in 20Mg at 3.70${}^{+0.02}_{-0.20}$ MeV and around 5.37 MeV from deuteron inelastic scattering is reported. The new data present a challenge for different nuclear forces from chiral effective field theory and ab initio calculations in the VS-IMSRG and CCEI frameworks since the results underpredict the observed resonance energy, although ground state binding energies were explained. This first systematic study shows that some of the different force prescriptions exhibit larger variance in the predicted energy, while the predictions from the two many body methods are similar using the same starting forces. The spin of the first excited state is determined to be 2+ and a neutron quadrupole deformation parameter $\beta_{n}$ = 0.46$\pm$0.21 is found from its differential cross section. This large deformation provides first signature of possible weakening of the $N$ = 8 shell closure at the proton drip-line. The data will therefore motivate further experimental and theoretical studies to explain the observed features.

The authors express sincere thanks to the TRIUMF beam delivery team. The support from NSERC, Canada Foundation for Innovation and the Nova Scotia Research and Innovation Trust is gratefully acknowledged. TRIUMF receives funding via a contribution through the National Research Council Canada. The support from RCNP for the target is gratefully acknowledged. It was partly supported by the grant-in-aid program of the Japanese government under the contract number 23224008 and 14J03935. We thank J. Simonis, K. Hebeler, and A. Schwenk for providing the EM interaction 3N matrix elements used in this work and for valuable discussions. Computations were performed with an allocation of computing resources at the Jülich Supercomputing Center (JURECA). This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Number DE-SC0018223 (SciDAC-4 NUCLEI) and the Field Work Proposals ERKBP57 and ERKBP72 at Oak Ridge National Laboratory (ORNL). This research used resources of the Oak Ridge Leadership Computing Facility located at ORNL, which is supported by the Office of Science of the Department of Energy under Contract No. DE- AC05-00OR22725. J.E. gratefully acknowledges financial support from the German Academic Exchange Service (DAAD Postdoc program).