First observation of unbound $^{11}$O, the mirror of the halo nucleus $^{11}$Li

TL;DR

This study reports the first experimental observation of the unbound proton-rich nucleus $^{11}$O, revealing its resonant states and their complex behavior, and compares it to its neutron-rich mirror $^{11}$Li.

Contribution

First experimental investigation of $^{11}$O, providing insights into its resonant states and isospin symmetry with $^{11}$Li using a Gamow coupled-channel approach.

Findings

Observed a broad peak of ~3 MeV in $^{11}$O decay spectrum.

Identified contributions from four low-lying resonant states with specific spins and parities.

Found strong, non-monotonic dependence of state properties on potential depth due to a broad threshold resonance.

Abstract

The structure of the extremely proton-rich nucleus $^{11}_{~8}$O$_3$, the mirror of the two-neutron halo nucleus $^{11}_{~3}$Li$_8$, has been studied experimentally for the first time. Following two-neutron knockout reactions with a $^{13}$O beam, the $^{11}$O decay products were detected after two-proton emission and used to construct an invariant-mass spectrum. A broad peak of width $\sim$3\,MeV was observed. Within the Gamow coupled-channel approach, it was concluded that this peak is a multiplet with contributions from the four-lowest $^{11}$O resonant states: $J^{\pi}$=3/2$^-_1$, 3/2$^-_2$, 5/2$^+_1$, and 5/2$^+_2$. The widths and configurations of these states show strong, non-monotonic dependencies on the depth of the $p$-$^9$C potential. This unusual behavior is due to the presence of a broad threshold resonant state in $^{10}$N, which is an analog of the virtual state in $^{10}$Li in the presence of the Coulomb potential. After optimizing the model to the data, only a moderate isospin asymmetry between ground states of $^{11}$O and $^{11}$Li was found.

Full text

Introduction.– There is increasing interest in nuclei with large differences in their number of protons and neutrons as these can have unusual structures such as spatially extended halos and low-energy intruder states. One of the most iconic of these exotic nuclei is 11Li, which processes a large two-neutron halo that gives it a physical size similar to the much heavier 208Pb nucleus Tanihata (1996). Borromean nuclei such as 11Li are loosely bound three-body systems (9Li core +2$n$) where there is no bound two-body subsystem Jensen et al. (2004). In 11Li, the two valence neutrons have roughly similar probabilities of occupying the $(s_{1/2})^{2}$ and $(p_{1/2})^{2}$ configurations, with the former largely accounting for the halo Hagino and Sagawa (2005). The $n$+9Li subsystem is also interesting as it may have an antibound, or virtual, state that could have important consequences for the halo structure of 11Li Thompson and Zhukov (1994); Betan et al. (2004); Michel et al. (2006); Simon et al. (2007); Aksyutina et al. (2008).

Mirror nuclei, with interchanged numbers of protons and neutrons, are expected to have similar nuclear structure due to isospin symmetry. In the case of 11Li, its mirror partner is the extremely proton-rich 11O nucleus located beyond the proton drip line which has not been observed until this work. With 8 protons and only 3 neutrons, its nearest particle-bound neighbor is 13O, two neutrons away. Both valence protons in 11O are unbound making its ground state (g.s.) a two-proton (2$p$) emitter, similar to 12O Jager et al. (2012).

The presence of unbound nucleons presents an appreciable challenge for nuclear theory Michel et al. (2009, 2010a). The interaction between localized shell-model states and the continuum has been shown to lead to a number of interesting properties including clusterization Okołowicz et al. (2012) and the breaking of mirror symmetry due to the Thomas-Ehrman effect Grigorenko et al. (2002); Michel et al. (2010b). There are predictions that the unbound 12O neighbor, has tens of percent more $(s_{1/2})^{2}$ occupation than its bound mirror 12Be Grigorenko et al. (2002). Based on the extrapolation of the quadratic isobaric multiplet mass equation (IMME) fit to the three neutron-rich members of the $A=11$ sextet, the g.s. of 11O should be unbound by 3.21(84) MeV Charity et al. (2012), significantly more than 12O ($\sim$1.7 MeV). Therefore, one might expect the effect of the continuum coupling may lead to even larger mirror symmetry breaking. The presence of broad threshold resonant states in the $p$+9C scattering channel could complicate these naïve expectations. Thus to understand the role of continuum couplings on the structure of 11O we have carried out both experimental and theoretical studies of this nucleus.

Experiment.– We produced 11,12O from one and two-neutron knockout reactions with a 13O beam. Only a few results for 12O will be presented in this work. The experiment was performed at the National Superconducting Cyclotron Laboratory at Michigan State University, which provided an $E/A=150$ MeV 16O primary beam. This beam bombarded a 9Be target and the A1900 magnetic separator selected out $E/A$=69.5-MeV 13O fragments with a purity of only 10%. The beam was then sent into an electromagnetic time-of-flight filter, the Radio Frequency Fragment Separator Bazin et al. (2009), and emerged with a purity of 80%. This secondary beam impinged on a 1-mm-thick 9Be target and the charged reaction products were detected 85 cm further downstream in the High Resolution Array (HiRA) Wallace et al. (2007) which consisted of 14 $\Delta$E-E [Si-CsI(Tl)] telescopes. This array subtended polar angles from $\theta_{lab.}$=2.1∘ to 12.4∘ in an arrangement almost identical to that in Charity et al. (2011); Brown et al. (2014, 2017).

Energy calibrations of the CsI(Tl) $E$ detectors were achieved with cocktail beams including $E/A=82.9$ MeV 9C fragments and $E=80$ MeV protons. Lower-energy calibration points were obtained using 2.2, 4.9, and 9.6 mm-thick Al degraders. Monte Carlo simulations Brown et al. (2014, 2017) were used to determine the invariant-mass resolution. The accuracy of these simulations, and that of the extracted centroid energies, were verified by studying the invariant mass of well-known levels, including the $J^{\pi}$=2${}^{+}_{1}$ and 2${}^{-}_{1}$ excited states of 12N, which decay via the $p$+11C channel Jager et al. (2012).

Figure 1 shows the 11O total 2$p$-decay energy ($Q_{2p}$) spectrum reconstructed with the invariant-mass method from detected 2$p$+9C events (a) before and (b) after background removal. This background results from fragments with undergo a nuclear reaction in a CsI(Tl) crystal, producing a smaller light output and are misidentified as a neighboring lighter isotope. From the calibration beams, we find 0.5% of the 11C and 10C fragments were misidentified as 9C. The two background curves in Fig. 1(a) were determined from 2$p$+10,11C coincidence events that were analyzed as if the detected carbon fragment was 9C.

The background-subtracted spectrum displays a broad structure of peak energy $\sim$4.5 MeV, which is higher than the IMME extrapolation for the g.s. of 3.21(84) MeV. To estimate the width of this structure, we fit it assuming a diproton decay in the $R$-matrix prescription Barker (2001), and incorporated the experimental resolution via the Monte Carlo simulations. However, we could not obtain a wide enough peak with the standard assumption for the diproton lineshape based on the $p$-$p$ phase shift. This already casts doubt that the peak is a singlet. Instead to obtain larger widths, we took the lineshape as a delta function centered at $E_{pp}$=0. The fitted intrinsic width was 3.4 MeV which is large compared to the experimental resolution (FWHM=0.45 MeV at $Q_{2p}$=4.5 MeV).

Theory.– To describe the spectra and two-proton ($2p$) decay of 11,12O, we utilized the three-body core+nucleon+nucleon Gamow coupled-channel (GCC) approach of Refs. Wang et al. (2017); Wang and Nazarewicz (2018) and we refer to these papers and to the review on the complex-energy shell model Michel et al. (2009) for technical details and basic concepts. The core (9,10C) is chosen as a deformed rotor, which can reasonably reproduce the intruder state and allows the pair of nucleons to couple to the collective states of the core. The wave function is constructed in Jacobi coordinates with a complex-energy basis, which can give exact asymptotic behavior of the wave functions and treats structure and reaction aspects on the same footing.

For the nuclear two-body interaction between valence nucleons, we took the finite-range Minnesota force Thompson et al. (1977) augmented by their Coulomb interaction. The effective core-valence potential has been taken in a deformed Woods-Saxon (WS) form including the spherical spin-orbit term Cwiok et al. (1987). The Coulomb core-proton potential is calculated assuming the core charge is uniformly distributed inside its deformed nuclear surface  Cwiok et al. (1987).

To analyze Thomas-Ehrman effects Thomas (1951); Ehrman (1951); Thomas (1952); Auerbach and Vinh Mau (2000); Michel et al. (2010b), the mirror nuclei with two valence neutrons, e.g., 11Li and 12Be, have been studied in a similar way. The deformed core is described by the quadrupole deformation $\beta_{2}$, and the couplings to the low-lying rotational states are included. The core rotational energies are taken from Ref. ENS . In the coupled-channel calculations, we included the g.s. band of the even-$A$ core with $J\leq j_{c}^{\rm max}=4^{+}$ and the odd-$A$ core with $J\leq j_{c}^{\rm max}=11/2^{-}$, respectively. According to previous work Wang and Nazarewicz (2018), the higher-lying rotational states have little influence on the final energy spectrum and g.s. mass.

Apart from the depth, the other parameters of the core-valence potential were optimized to fit the 1/2${}_{1}^{+}$, 1/2${}_{1}^{-}$, and 5/2${}_{1}^{+}$ levels of 11N ENS using the quasi-Newton method. At the $\chi^{2}$-minimum, the root-mean-square (rms) deviation is 197 keV. The resulting values are: spin-orbit potential $V_{\rm s.o.}=15.09$ MeV, diffuseness $a=0.7$ fm, the WS (and charge) radius $R=2.3$ fm, and the quadrupole deformation $\beta_{2}=0.52$; they are similar to those in Ref. Fossez et al. (2016), which can reasonably reproduce the intruder state of 11Be. The depth $V_{0}$ used in our 11O analysis, was adjusted to fit the 1${}_{1}^{-}$ and 2${}_{1}^{-}$ states of the core+nucleon system 10N with the rms error of 143 keV.

The GCC configurations can be described both in the original Jacobi coordinates $(S,\ell_{x},\ell_{y})$ and the cluster orbital shell model coordinates $(j_{1},j_{2})$, where $S$ is the total spin of the valence nucleons and $\ell_{x}$ and $\ell_{y}$ are, respectively, the orbital angular momenta of two-protons about their center of mass and of this center of mass about the core. The calculations have been carried out in the model space of $\max(\ell_{x},\ell_{y})\leq 7$ with the maximal hyperspherical quantum number $K_{\rm max}=20$. For the hyperradial part, we used the Berggren basis for the $K\leq 6$ channels and the harmonic oscillator basis with the oscillator length $b=1.75$ fm and $N_{\rm max}=40$ for the higher-angular-momentum channels. The complex-momentum contour of the Berggren basis is defined as: $k=0\rightarrow 0.4-0.2i\rightarrow 0.6\rightarrow 2\rightarrow 4\rightarrow 8$ (all in fm*-1*), with each segment discretized with 60 points. To study antibound states and broad resonant states in the core-valence potential, we used the deformed complex-momentum contour as in Refs. Betan et al. (2004); Michel et al. (2006).

Discussion.– To benchmark the theory, we calculated the 12O g.s. The experimental g.s. corresponds to a Breit-Wigner resonance having a centroid of $Q_{2p}$=1.688(29) MeV and an intrinsic width of $\Gamma$=51(19) keV. This experimental $Q_{2p}$ value is slightly larger than the value of 1.638(24) MeV from a previous invariant-mass study Jager et al. (2012), but still smaller than the values of 1.783(48), 1.760(24), and 1.740(22) MeV obtained from the 12C($\pi^{+}$,$\pi^{-}$) reactions Fortune and Sherr (2013). Our calculations give $Q_{2p}$=1.97 MeV and $\Gamma$=120 keV, which is in reasonable agreement with experiment. If the depth $V_{0}$ of the WS potential is adjusted to reproduce the experimental $Q_{2p}$ value, the theoretical width is now 18${}^{+4}_{-3}$ keV, closer to our extracted value. The $(s_{1/2})^{2}$ configuration is predicted to account for 35% of the strength compared to 20% for the mirror nucleus 12Be calculated with the same parameters. This level of mirror symmetry violation is roughly in accord with the predictions of Ref. Grigorenko et al. (2002).

From the initial estimate of $V_{0}$, we obtain $Q_{2p}$=3.17 MeV, $\Gamma$=0.86 MeV for 11Og.s., which cannot explain the experimental peak. To investigate whether this peak can be a singlet, we have varied the depth of the WS potential to the analyze how the decay width changes. If $Q_{2p}$ is set at 4.55 MeV, i.e., close to the maximum of the experimental peak, the decay width is still only 1.29 MeV with 67.7% of ($K$, $S$, $\ell_{x}$) = (0, 0, 0) configuration. As this configuration is largely responsible for the decay rate Grigorenko et al. (2002), then we can obtain a maximum value of $\Gamma$=1.29/0.67$\approx$1.9 MeV. Another rough estimate is obtained by assuming there is no valence-nucleon interaction and recoil term. Based on the three-body model, the g.s. energy and decay width of 11O would be two times as large as that of 11N. Hence, the decay energy $Q_{2p}$ should be around 4 MeV with a width around 1.5 MeV, which is still significantly less than the experimental value. Thus, we conclude that the observed peak must contain multiple components.

We next consider the possibility that the experimental peak is a doublet of the two predicted 3/2- states. Attempts to reproduce it by varying the depth of the WS potential to change $Q_{2p}$ and using the calculated line shapes to fit the data failed as the best-fit spectrum was still too narrow. Thus we are forced to also include contributions from the two 5/2+ states which would require the two knocked-out neutrons to come from different shells to conserve parity. An analogous state has been observed in the mirror reaction, i.e., in two-proton knockout from 13B at a similar bombarding energy, the first-excited state of 11Li was observed Smith et al. (2016) which has positive parity Kanungo et al. (2015); Tanaka et al. (2017). This state is consistent with the 5/2${}^{+}_{1}$ 11Li state obtained in our calculations. The energies $Q_{2p}$ (and widths) of the four lowest-lying resonant states in 11O obtained with $V_{0}$ optimized to the observed energy spectrum are: 4.16 (1.30) MeV for $3/2^{-}_{1}$; 4.65 (1.06) MeV for $5/2^{+}_{1}$; 4.85 (1.33) MeV for $3/2^{-}_{2}$; and 6.28 (1.96) MeV for $5/2^{+}_{2}$. The statistical uncertainty in $V_{0}$ obtained from a $\chi^{2}$ analysis implies uncertainties of about 5 keV on the centroids and 2 keV on the widths. Based on our past experience, the centroids also have an additional $\pm$10 keV systematic uncertainty.

The $Q_{2p}$ value for the $3/2^{-}_{1}$ state is only 1.1$\sigma$ from the IMME extrapolation Charity et al. (2012). Figure 1(b) shows the best fit with the contributions from these four levels. These line shapes have been modified to incorporate the experimental resolution. With a smooth, rather flat background contribution included, the data are well described. The 3/2${}^{-}_{1}$ and 5/2${}^{+}_{2}$ peaks make up 39% and 32% of the fitted yield, respectively, with the remainder attributed to the 3/2${}^{-}_{2}$ and 5/2${}^{+}_{1}$ levels, which have similar centroids and widths and thus cannot be disentangled.

The dependence of the predicted widths on the depth $V_{0}$, and hence $Q_{2p}$, is complicated by the presence of broad threshold resonant states in 10N, which affect the behavior of the $\ell=0$ single-particle channel. Figure 2(a) shows the $2p$ partial decay widths of the 3/2- resonant states of 11O as a function of $Q_{2p}$. The predicted decay width of the 3/2${}^{-}_{1}$ g.s. increases rapidly with $Q_{2p}$ below the Coulomb barrier ($\sim$3.2 MeV); this is accompanied by a rapid configuration change, see Fig. 2(b). The wave function of this state is dominated by the single $(S,\ell_{x},\ell_{y})=(000)$ Jacobi-coordinate component. For $Q_{2p}<2.2$ MeV, the weight of the ($s_{1/2})^{2}$ shell-model component is around 20%, which is similar to predictions for the mirror nucleus 11Li. As $Q_{2p}$ gets larger, the $(s_{1/2})^{2}$ amplitude rapidly increases. Close to the Coulomb barrier, this state becomes a pure $(s_{1/2})^{2}$ configuration. At energies above the barrier, the wave function has a small amplitude inside the nuclear volume and can no longer be associated with an outgoing solution; it dissolves into the scattering continuum. Interestingly, as seen in Fig. 2(a), a second branch of the 3/2${}^{-}_{1}$ solution appears at higher $Q_{2p}$ values. This behavior is attributed to the presence of a broad resonant state in the $\ell=0$ $p$-9C scattering channel, which is an analog of the antibound state of 10Li. When steadily increasing the Coulomb interaction from zero ($n+^{9}$Li) to the full $p+^{9}$C value at $V_{0}=-52.17$ MeV ($Q_{2p}=4.13$ MeV), this resonant pole evolves in the complex-$k$ plane from the antibound state in 10Li with $k=-0.222i$ fm*-1* ($E=-1.022$ MeV), passing the region of subthreshold resonances Re$(E)<0$ and $\Gamma>0$, located below the $-45^{\circ}$ line in the momentum plane Sofianos et al. (1997); Mukhamedzhanov et al. (2010), and eventually becoming the broad $s$-wave threshold resonant state in 10N with $k=0.252-0.213i$ fm*-1* ($E=0.38$ MeV, $\Gamma=4.45$ MeV), see Ref. Efros and Bang (1999); Lovas et al. (2002); Csótó (2002); Mao et al. (2018) for more discussion. This antibound-state analog is present in the broad range of $Q_{2p}$ values, and is the source of the discontinuity between the two branches for the 3/2${}^{-}_{1}$ state when it gets close to the $-45^{\circ}$ line.

The second 3/2${}^{-}_{1}$ solution shows a similar trend to the first one, with the amplitude of the $(s_{1/2})^{2}$ configuration increasing with $Q_{2p}$. Our fitted value of $Q_{2p}$ for the g.s. corresponds to the second solution in Fig. 2 and contains 29% of the $(s_{1/2})^{2}$ configuration compared a similar value of 25% for the mirror nucleus 11Li. However at other values of $Q_{2p}$, the breaking of isospin symmetry would have been very substantial. The 3/2${}^{-}_{2}$ state exhibits a similar discontinuity as the 3/2${}^{-}_{1}$ resonant solution; the dependence of the 3/2${}^{-}_{2}$ $(s_{1/2})^{2}$ strength on $Q_{2p}$ is roughly inverted to that for 3/2${}^{-}_{1}$ state in Fig. 2(b), with excited-state component dropping to zero when the ground-state approaches the pure $(s_{1/2})^{2}$ configuration.

One can see the Thomas-Ehrman effect directly in the wave functions of the valence nucleons. Figure 3 shows the predicted two-nucleon density distributions in Jacobi coordinates (defined in Ref. Wang et al. (2017)) for the 3/2${}^{-}_{1}$, 5/2${}^{+}_{1}$, and 3/2${}^{-}_{2}$ resonant states of 11Li and 11O obtained with our optimized value of $V_{0}$ for the latter. The two 3/2- states show strong correlations between the valence nucleons with either diproton or dineutron characteristics. As expected, the diproton peak in the unbound 11O is slightly less localized than that of the dineutron configuration in the bound mirror halo system. However, the secondary peak strength for cigarlike arrangements is significantly reduced in 11O. The two-nucleon correlations for the 5/2${}^{+}_{1}$ state are rather weak, with the maximum density occurring for large separations between the two valence nucleons. Again, the 5/2${}^{+}_{1}$ wave function is more extended spatially in 11O than in 11Li. This resonant state is dominated by the $(s_{1/2},p_{1/2})$ component, while the 5/2${}^{+}_{2}$ level in 11O is dominated by the $(s_{1/2},p_{3/2})$ component.

Conclusions.– The proton-unstable isotope 11O has been observed for the first time. It was produced by two-neutron knockout reactions from a 13O beam. The invariant-mass spectrum of its 2$p$+9C decay products measured with the HiRA detector contained a single broad peak with a width of 3.4 MeV. The low-energy structure of 11O was also studied theoretically with the Gamow coupled-channel approach which suggests that the observed peak is most likely a multiplet. We obtained an excellent fit to this structure with contributions from the four-lowest excited resonant GCC states ($J^{\pi}$=3/2${}^{-}_{1}$,5/2${}^{+}_{1}$,3/2${}^{-}_{2}$,5/2${}^{+}_{2}$).

The predicted width of the 3/2${}^{-}_{1}$ g.s. shows complicated variation with the depth of the confining potential due to the presence of broad resonant states in the $p$-9C scattering $\ell=0$ channel. With our fitted depth, the g.s. configuration was found to be similar to its mirror system 11Li. However, significantly different configurations were predicted for other values of the depth. These results demonstrate the importance of the coupling to the continuum for states beyond the drip lines and the role that $\ell=0$ near-threshold resonant states can play in constructing the many-body wave functions. We also studied the Thomas-Ehrman effect directly in the wave functions of valence nucleons in the mirror 11O-11Li pair. According to our calculations, the strength in the diproton (cigar) configuration is relatively stronger (weaker) in the g.s. of 11O compared to the situation in 11Li.

Acknowledgments.– This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under award numbers DE-FG02-87ER-40316, DE-FG02-04ER-41320,DE-SC0014552, DE-SC0013365 (Michigan State University), DE-SC0018083 (NUCLEI SciDAC-4 collaboration), and DE-SC0009971 (CUSTIPEN: China-U.S. Theory Institute for Physics with Exotic Nuclei); and the National Science foundation under grant PHY-156556. J.M. was supported by a Department of Energy National Nuclear Security Administration Steward Science Graduate Fellowship under cooperative agreement number DE-NA0002135.