Structures and decay properties of extremely proton-rich nuclei $^{11,12}$O

TL;DR

This study uses a three-body Gamow coupled-channel approach to analyze the structure and decay properties of the proton-rich nuclei $^{11}$O and $^{12}$O, revealing the influence of continuum effects and mirror symmetry on their resonant states.

Contribution

It introduces a detailed three-body model with realistic interactions to investigate the decay dynamics and continuum effects in $^{11}$O and $^{12}$O, including their mirror nuclei.

Findings

Broad resonant structures in $^{11}$O consistent with multiple resonances.

Decay dynamics show competition between direct and democratic $2p$ emission.

The structure of $^{10}$N influences the low-lying states of $^{11}$O.

Abstract

${\rm \bf Background:}$ The recent observation of the unbound nucleus $^{11}$O offers the unique possibility to study how the structure and dynamics of two-proton ($2p$) decay is affected by the removal of one neutron from $^{12}$O, and provides important information on the Thomas-Ehrman effect in the mirror pairs $^{11}_{~8}$O$_3$-$^{11}_{~3}$Li$_8$ and $^{12}_{~8}$O$_4$-$^{12}_{~4}$Be$_8$, which involve the $2p$ emitters $^{11}$O and $^{12}$O. ${\rm \bf Purpose:}$ We investigate how continuum effects impact the structure and decay properties of $^{11}$O and $^{12}$O, and their mirror partners. ${\rm \bf Methods:}$ We solve the three-body core-nucleon-nucleon problem using the Gamow coupled-channel (GCC) method. The GCC Hamiltonian employs a realistic finite-range valence nucleon-nucleon interaction and the deformed cores of $^{9,10}$C, $^{9}$Li, and $^{10}$Be. ${\rm \bf Results:}$ We calculate the energy spectra and decay widths of $^{11}$O and $^{12}$O as well as those of their mirror nuclei. In particular, we investigate the dynamics of the $2p$ decay in the ground state of $^{12}$O by analyzing the evolution of the $2p$ configuration of the emitted protons as well as their angular correlations in the coordinate space. We also show how the analytic structure of the resonant states of $^{10}$Li and $^{10}$N impacts the low-lying states of $^{11}$Li and $^{11}$O. ${\rm \bf Conclusions:}$ We demonstrate that, in both nuclei $^{11}$O and $^{12}$O, there is a competition between direct and "democratic" $2p$ ground-state emission. The broad structure observed in $^{11}$O is consistent with four broad resonances, with the predicted $3/2^-_1$ ground state strongly influenced by the broad threshold resonant state in $^{10}$N, which is an isobaric analog of the antibound (or virtual) state in $^{10}$Li.

Figures

Tables

Table 1. Table 1: Predicted energies and widths (both in MeV) for low-lying states in 11,12 O and their mirror systems. For 11 Li, the parameters of the core-nucleon potential are taken from Ref.  Hagino and Sagawa ( 2005a ) . For the remaining nuclei, the optimized parameters are used with the depth of the core-nucleon potential being readjusted to reproduce the g.s. energy of each nucleus. Also listed are the dominant Jacobi ( S 𝑆 S , ℓ x subscript ℓ 𝑥 \ell_{x} , ℓ y subscript ℓ 𝑦 \ell_{y} ) and COSM ( j 1 subscript 𝑗 1 j_{1} , j 2 subscript 𝑗 2 j_{2} ) configurations.

Nucleus	$J^{\pi }$	$E\left({\Gamma }_{2p}\right)$	($S$, ${\ell }_x$, ${\ell }_y$)	($j_1$, $j_2$)
11Li	$3/2_1^{-}$	$-0.242$	61% (0, 0, 0)	63% ($p_{1/2}$, $p_{1/2}$)
			33% (1, 1, 1)	25% ($s_{1/2}$, $s_{1/2}$)
	$5/2_1^{+}$	0.664(0.258)	42%(0, 0, 1)	94% ($s_{1/2}$, $p_{1/2}$)
			40% (1, 1, 0)	2% ($s_{1/2}$, $p_{3/2}$)
	$3/2_1^{-}$	1.318(1.242)	58% (0, 0, 0)	61% ($p_{1/2}$, $p_{1/2}$)
			33% (1, 1, 1)	35% ($s_{1/2}$, $s_{1/2}$)
11O	$3/2_1^{-}$	4.158(1.296)	66% (0, 0, 0)	54% ($p_{1/2}$, $p_{1/2}$)
			33% (1, 1, 1)	29 % ($s_{1/2}$, $s_{1/2}$)
				6 % ($s_{1/2}$, $d_{5/2}$)
				3% ($d_{5/2}$, $d_{5/2}$)
	$5/2_1^{+}$	4.652(1.055)	43%(0, 0, 1)	84% ($s_{1/2}$, $p_{1/2}$)
			33% (1, 1, 0)	3% ($p_{1/2}$, $d_{5/2}$)
			11% (1, 1, 2)	3% ($s_{1/2}$, $p_{3/2}$)
			3% (0, 2, 1)
	$3/2_2^{-}$	4.850(1.334)	70% (0, 0, 0)	43% ($s_{1/2}$, $s_{1/2}$)
			28% (1, 1, 1)	42% ($p_{1/2}$, $p_{1/2}$)
				3% ($s_{1/2}$, $d_{5/2}$)
	$5/2_2^{+}$	6.283(1.956)	47% (0, 0, 1)	87% ($s_{1/2}$, $p_{3/2}$)
			43% (1, 1, 0)	4% ($p_{1/2}$, $d_{5/2}$)
			4% (1, 1, 2)	2% ($s_{1/2}$, $p_{1/2}$)
12Be	$0_1^{+}$	$-3.672$	54% (0, 0, 0)	35% ($p_{1/2}$, $p_{1/2}$)
			30% (1, 1, 1)	25% ($d_{5/2}$, $d_{5/2}$)
			6% (0, 0, 2)	20% ($s_{1/2}$, $s_{1/2}$)
			4% (0, 2, 0)	14% ($s_{1/2}$, $d_{5/2}$)
	$2_1^{+}$	$-1.344$	29% (0, 0, 2)	49% ($s_{1/2}$, $d_{5/2}$)
			21% (1, 1, 1)	30% ($d_{5/2}$, $d_{5/2}$)
			20% (0, 0, 0)	9% ($s_{1/2}$, $s_{1/2}$)
			11% (0, 2, 0)
			7% (1, 1, 3)
			6% (1, 3, 1)
	$0_2^{+}$	$-1.129$	42% (1, 1, 1)	57% ($p_{1/2}$, $p_{1/2}$)
			37% (0, 0, 0)	16% ($s_{1/2}$, $s_{1/2}$)
			8%(0, 0, 2)	15% ($s_{1/2}$, $d_{5/2}$)
				3% ($d_{5/2}$, $d_{5/2}$)
	$1_1^{-}$	$-0.983$	31% (0, 0, 1)	78% ($s_{1/2}$, $p_{1/2}$)
			31% (1, 1, 0)	16% ($p_{1/2}$, $d_{5/2}$)
			15% (1, 1, 2)
			6% (0, 0, 3)
			5% (0, 2, 1)
	$2_2^{+}$	1.125(0.091)	35% (1, 1, 1)	44% ($p_{1/2}$, $p_{1/2}$)
			25% (0, 0, 0)	39% ($p_{1/2}$, $p_{3/2}$)
			21% (0, 0, 2)	6% ($s_{1/2}$, $d_{5/2}$)
12O	$0_1^{+}$	1.688(0.018)	65% (0, 0, 0)	36% ($s_{1/2}$, $s_{1/2}$)
			21% (1, 1, 1)	25% ($p_{1/2}$, $p_{1/2}$)
				14% ($d_{5/2}$, $d_{5/2}$)
				13% ($s_{1/2}$, $d_{5/2}$)
	$0_2^{+}$	3.162(0.818)	90% (0, 0, 0)	71% ($s_{1/2}$, $s_{1/2}$)
			7% (1, 1, 1)	23% ($p_{1/2}$, $p_{1/2}$)
				3% ($s_{1/2}$, $d_{5/2}$)
	$1_1^{-}$	3.256(0.516)	43% (0, 0, 1)	90% ($s_{1/2}$, $p_{1/2}$)
			36% (1, 1, 0)	4% ($p_{1/2}$, $d_{5/2}$)
			8% (1, 1, 2)	3% ($s_{1/2}$, $p_{3/2}$)
	$2_1^{+}$	3.802(0.132)	32% (0, 0, 2)	65% ($s_{1/2}$, $d_{5/2}$)
			20% (1, 1, 1)	23% ($d_{5/2}$, $d_{5/2}$)
			17% (0, 0, 0)	11% ($s_{1/2}$, $s_{1/2}$)
	$2_2^{+}$	5.150(1.027)	19% (1, 1, 3)	94% ($p_{1/2}$, $p_{3/2}$)
			17% (1, 3, 1)	1% ($p_{1/2}$, $p_{1/2}$)
			16% (1, 1, 1)
			12% (0, 2, 0)
			11% (0, 0, 2)

Equations

$$\hat{H}=\sum^{3}_{i=c,p_{1},p_{2}}\frac{\hat{\mbox{\boldmath$p$}}^{2}_{i}}{2m_{i}}+\sum^{3}_{i>j=1}V_{ij}(\mbox{\boldmath$r$}_{ij})+\hat{H_{c}}-\hat{T}_{\rm c.m.},$$

Full text

The recent observation of the unbound nucleus 11O offers the unique possibility to study how the structure and dynamics of two-proton ($2p$) decay is affected by the removal of one neutron from 12O, and provides important information on the Thomas-Ehrman effect in the mirror pairs ${}^{11}_{~{}8}$O3-${}^{11}_{~{}3}$Li8 and ${}^{12}_{~{}8}$O4-${}^{12}_{~{}4}$Be8, which involve the $2p$ emitters 11O and 12O.

Purpose

We investigate how continuum effects impact the structure and decay properties of 11O and 12O, and their mirror partners.

Methods

We solve the three-body core-nucleon-nucleon problem using the Gamow coupled-channel (GCC) method. The GCC Hamiltonian employs a realistic finite-range valence nucleon-nucleon interaction and the deformed cores of 9,10C, 9Li, and 10Be.

Results

We calculate the energy spectra and decay widths of 11O and 12O as well as those of their mirror nuclei. In particular, we investigate the dynamics of the $2p$ decay in the ground state of 12O by analyzing the evolution of the $2p$ configuration of the emitted protons as well as their angular correlations in the coordinate space. We also show how the analytic structure of the resonant states of 10Li and 10N impacts the low-lying states of 11Li and 11O.

Conclusions

We demonstrate that, in both nuclei 11O and 12O, there is a competition between direct and “democratic” $2p$ ground-state emission. The broad structure observed in 11O is consistent with four broad resonances, with the predicted $3/2^{-}_{1}$ ground state strongly influenced by the broad threshold resonant state in 10N, which is an isobaric analog of the antibound (or virtual) state in 10Li.

I Introduction

Weakly bound and unbound drip-line nuclei having a large proton-to-neutron imbalance are susceptible to clustering effects due to the presence of low-lying decay channels Ikeda et al. (1968); Jonson (2004); von Oertzen et al. (2006); Freer (2007); Okołowicz et al. (2012, 2013). Wave functions of such systems often “align” with the nearby threshold and are expected to have large overlaps with the corresponding decay channels. Many examples of threshold phenomena can be found in dripline nuclei Ikeda et al. (2010); Papadimitriou et al. (2011); Spyrou et al. (2012); Lovell et al. (2017); Casal (2018); Hagino and Sagawa (2016a, b); Brown et al. (2015); Miernik et al. (2007) exhibiting dineutron- and diproton-type correlations, as well as exotic $2n$ and $2p$ decay modes Dobaczewski et al. (2007); Forssén et al. (2013); Pfützner et al. (2012); Pfützner (2013); Thoennessen (2004); Blank and Płoszajczak (2008); Grigorenko et al. (2011); Olsen et al. (2013); Kohley et al. (2013).

To gain insight into the nature of threshold effects, it is useful to study pairs of mirror nuclei whose energy spectra and structure must be identical assuming an exact isospin symmetry. In reality, of course, differences are always present due to electromagnetic effects (primarily Coulomb interaction), which also result in asymmetries between proton and neutron thresholds and different asymptotic behavior of proton and neutron wave functions, both manifested through the Thomas-Ehrman effect Thomas (1951); Ehrman (1951); Thomas (1952); Auerbach and Vinh Mau (2000); Grigorenko et al. (2002a); Michel et al. (2010).

The recent observation of the unbound nucleus 11O Webb et al. (2019a) provides several unique opportunities in that regard. First, 11O is a $2p$ emitter and the mirror to the $2n$ Borromean halo system 11Li, which allows for the study of the Thomas-Ehrman effect in the extreme case involving proton-unbound and neutron halo systems. Second, 12O is also a $2p$ emitter Kryger et al. (1995); Jager et al. (2012); Webb et al. (2019b) and the mirror of the bound nucleus 12Be, which is deformed and exhibits cluster effects Freer et al. (1999); Kanada-En’yo and Horiuchi (2003); Ito et al. (2008); Yang et al. (2015). At a deeper level, new discoveries Webb et al. (2019a, b) provide important insights on the continuum couplings in $2p$ emitters and $2n$ halos. The 10,11N subsystems of 11,12O and their mirror nuclei 10Li and 11Be all present interesting continuum features. For instance, the nucleus 10Li has a antibound, or virtual, state Thompson and Zhukov (1994); Betan et al. (2004); Michel et al. (2006); Simon et al. (2007); Aksyutina et al. (2008) whose isobaric analog state in 10N is a broad threshold resonance Aoyama et al. (1997); Hooker et al. (2017); Webb et al. (2019a). The nucleus 11Be shows a celebrated ground-state (g.s.) parity inversion, which is also found in 11N Markenroth et al. (2000); Oliveira et al. (2000); Guimarães et al. (2003). Consequently, studies of mirror pairs 11O-11Li, 12O-12Be, 10N-10Li, and 11N-11Be, offer unique perspectives on near-threshold clustering phenomena in light exotic nuclei and physics of nuclear open quantum systems in general.

From a theoretical point of view, the $2p$ emitters 11O and 12O can be described as three-body systems made of two valence protons coupled to deformed cores of 9C and 10C, respectively. Their mirror partners 11Li and 12Be can be described in a similar way but with two valence neutrons instead. A key ingredient to describe all those systems is the treatment of the continuum space. This is achieved in the GCC method Wang et al. (2017); Wang and Nazarewicz (2018), which was recently used in Ref. Webb et al. (2019a) to interpret the first data on 11O.

The objective of this work is to shed light on the dynamics of $2p$ decay and on the Thomas-Ehrman effect in extreme mirror nuclei, by studying the structure of the mirror pairs 11O-11Li and 12O-12Be in a common three-body framework including continuum coupling effects. In particular, we investigate angular correlations and $2p$ decay dynamics of 11,12O, which exhibit unique three-body features as well as the presence of strong coupling between the structure and reaction aspects of the three body problem.

This article is organized as follows. Section II contains the description of the model used. In particular, it lays out the framework of the deformed GCC method and defines the configuration space employed. The results for 11,12O and their mirror partners are presented in Sec. III. Finally, Sec. IV contains the summary and outlook.

II THE Model

II.1 Gamow coupled-channel method

To describe the energy spectra and $2p$ decay of 11,12O, we use the three-body core+nucleon+nucleon GCC approach Wang et al. (2017); Wang and Nazarewicz (2018). The core (9,10C) is chosen as a deformed rigid rotor. This core can reproduce in a reasonable way the deformed intruder state containing the large $s_{1/2}$ component and allow the pair of nucleons to couple to the collective states of the core through a non-adiabatic rotational-coupling. The three-body Hamiltonian of GCC is defined as:

[TABLE]

where the second sum captures the pairwise interactions between the three clusters, $\hat{H_{c}}$ is the core Hamiltonian represented by the excitation energies of the core $E^{j_{c}\pi_{c}}$, and $\hat{T}_{\rm c.m.}$ stands for the center-of-mass term.

The wave function of the parent nucleus can be written as $\Psi^{J\pi}=\sum_{J_{p}\pi_{p}j_{c}\pi_{c}}\left[\Phi^{J_{p}\pi_{p}}\otimes\phi^{j_{c}\pi_{c}}\right]^{J\pi}$, where $\Phi^{J_{p}\pi_{p}}$ and $\phi^{j_{c}\pi_{c}}$ are the wave functions of the two valence protons and the core, respectively. The wave function of the valence protons $\Phi^{J_{p}\pi_{p}}$ is expressed in Jacobi coordinates and expanded using the Berggren basis Berggren (1968); Michel et al. (2009); Wang et al. (2017) which is defined in the complex-momentum $k$ space. Since the Berggren basis is a complete ensemble that includes bound, Gamow and scattering states, it provides the correct outgoing asymptotic behavior to describe the $2p$ decay, and effectively allows the treatment of nuclear structure and reactions on the same footing.

The antisymmetrization between core and valence protons is taken care of by eliminating the Pauli-forbidden states occupied by the core nucleons using the supersymmetric transformation method Thompson et al. (2000, 2004); Descouvemont et al. (2003), which introduces an auxiliary repulsive “Pauli core” in the original core-valence interaction. For simplicity, in this work, we only project out the spherical orbitals corresponding to the deformed levels occupied in the daughter nucleus.

II.2 Hamiltonian and model space

The nuclear two-body interaction between valence nucleons is represented by the finite-range Minnesota force with the parameters of Ref. Thompson et al. (1977), which is supplemented by the two-body Coulomb force in the proton space. 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 $Z_{c}e$ is uniformly distributed inside the deformed nuclear surface Cwiok et al. (1987).

The deformed cores 9C and 10C (of 11O and 12O, respectively) are represented by WS potentials with a quadrupole deformation $\beta_{2}$. The couplings to the low-lying rotational states are fully included in the present formalism. The core rotational energies are taken from experiment ENS . In the coupled-channel calculations, we included the ground-state (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 the previous work Wang and Nazarewicz (2018), the higher-lying rotational states have little influence on the final energy spectra. A similar treatment is used for a construction for the deformed cores 9Li and 10Be (of 11Li and 12Be, respectively).

Except for the WS depth $V_{0}$, the parameters of the core-valence potentials were optimized to reproduce the energy spectrum of 11N ENS using a particle-plus-rotor model including continuum couplings through the Berggren basis. The fitted parameters are: spin-orbit strength $V_{\rm s.o.}=15.09$ MeV, diffuseness $a=0.7$ fm, WS (and charge) radius $R=1.106A_{c}^{1/3}$ fm, and quadrupole deformation $\beta_{2}=0.52$. These values are similar to those of Ref. Fossez et al. (2016) that reproduce the intruder band in 11Be in a reasonable way. Finally, the WS depth was readjusted to reproduce the energy spectra of the core+nucleon systems 10N and 11N; these values were then used in predictions for 11O and 12O, as well as for the mirror nuclei 11Li and 12Be. For comparison, we have also used a different WS parametrization from Ref. Hagino and Sagawa (2005a) for the $A=11$ systems 11O and 11Li. In this case, the parameters are: $V_{0}=-47.50$ MeV ($-35.37$ MeV) for even (odd) orbital angular momentum $\ell$, $V_{\rm s.o.}=-0.1785V_{0}$, $a=0.67$ fm, $R=1.27A_{c}^{1/3}$ fm, and $\beta_{2}=0$. As the core+valence potential of Ref. Hagino and Sagawa (2005a) is spherical, different depths for different $\ell$-channels are used in order to describe the $2s_{1/2}$ intruder state in this region.

The GCC configurations can be expressed both in the original Jacobi coordinates $(S,\ell_{x},\ell_{y})$ and in the cluster orbital shell model (COSM) coordinates $(j_{1},j_{2})$, where $S$ is the total spin of the valence nucleons and $\ell_{x}$ is the orbital angular momentum of the proton pair with respect to their center of mass and $\ell_{y}$ is the pair’s orbital angular momentum with respect to the core. The calculations were carried out in a model space defined by $\max(\ell_{x},\ell_{y})\leq 7$ and for a maximal hyperspherical quantum number $K_{\rm max}=20$. In the hyperradial part, we used the Berggren basis for the $K\leq 6$ channels and the harmonic oscillator (HO) 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 by the path $k=0\rightarrow 0.4-0.2i\rightarrow 0.6\rightarrow 2\rightarrow 4\rightarrow 8$ (all in fm*-1*), with each segment discretized by 60 points (scattering states). In order to study antibound states and broad resonances in the core-valence potential, we used the deformed complex-momentum contour as in Refs. Betan et al. (2004); Michel et al. (2006).

III Results

III.1 12O and its isobaric analog

III.1.1 Spectra

Exotic $p$-shell nuclei with a large proton-neutron asymmetry tend to clusterize, which results in profound structural changes near the drip lines. In Fig. 1 we show the energy spectra of 12O, 11N, 12Be, and 11Li. As can be seen, the ordering of the lowest $1/2^{-}$ and $1/2^{+}$ levels in 11N has been reproduced. The calculated $2p$ decay energy $Q_{2p}$ of 12O is 1.973 MeV with a decay width of 120 keV while the recently measured energy is 1.688(29) MeV with a decay width of 51(19) keV Webb et al. (2019b).

The GCC calculations predict several 12O excited stated in the energy range explored experimentally Suzuki et al. (2016); Webb et al. (2019b). In particular, we predict an excited $J^{\pi}=1_{1}^{-}$ state located between the 0${}_{2}^{+}$ and the 2${}_{1}^{+}$ states. This sequence differs from the level ordering in the mirror system 12Be due to the large Thomas-Ehrman shift. The location of the calculated 1${}_{1}^{-}$ state corresponds to the shoulder in the measured invariant-mass spectrum of 12O Webb et al. (2019b). Because the width of the 1${}_{1}^{-}$ state is similar to that of the 0${}_{2}^{-}$ state, it might be hidden in the observed peaks attributed to 0${}_{2}^{+}$ and 2${}_{1}^{+}$ states.

The $2p$ decay energy $Q_{2p}$ can be controlled by varying the depth of the core-proton potential. Figure 2 shows the calculated decay width of the g.s. of 12O versus $Q_{2p}$. Since in the range of considered $Q_{2p}$ values the g.s. of 12O lies below the Coulomb barrier, the decay width increases almost exponentially with $Q_{2p}$. To compare with the results of Ref. Grigorenko et al. (2002b), we calculate the decay width at their reported value of $Q_{2p}=1.790$ MeV. The resulting decay width is 35.2 keV with a dominant configuration $(K,S,\ell_{x})=(0,0,0)$ at 43.3% of the wave function, which is slightly smaller than 66.6% reported in Ref. Grigorenko et al. (2002b). This difference is caused by different potential parameters, deformation, and configuration mixing of the excited states of the core. By taking the same $(K,S,\ell_{x})=(0,0,0)$ amplitude as in Ref. Grigorenko et al. (2002b), the decay width obtained in our model would be $35.2\times 66.6\%/43.3\%=54.1$ keV, which is in agreement with the value of 56.9 keV reported in Ref. Grigorenko et al. (2002b). Following this idea, one can estimate an upper limit of 81.3 keV for the decay width at $Q_{2p}$ = 1.790 MeV by assuming that the valence protons occupy a pure $(K,S,\ell_{x})=(0,0,0)$ state.

To compare with experiment Webb et al. (2019b), we slightly readjusted the depth of core-proton potential to reproduce the remeasured $Q_{2p}$ value of 12O. The resulting decay width (marked ‘GCC refit’ in Fig. 2) is 18${}^{+4}_{-3}$ keV, which is slightly less than the remeasured value. We believe the theoretical predication are quite reasonable considering detector resolution as well as the significant spread of the experimental results KeKelis et al. (1978); Kryger et al. (1995); Suzuki et al. (2009); Jager et al. (2012); Webb et al. (2019b).

Table 1 shows the dominant configurations in Jacobi and COSM coordinates. The wave functions of the mirror nuclei 12O and 12Be differ significantly due to the large Thomas-Ehrman effect. Namely, the contributions from the $\ell=0$ partial waves are systematically larger in the unbound 12O. This is in accord with the results of Ref. Grigorenko et al. (2002b).

III.1.2 Angular correlations

As seen in Fig. 1, the g.s. of 11N lies between those of 12O and 10C. This opens a possibility for the competition between the direct and sequential $2p$ decays in 12O. To illustrate how this affects the decay properties, we now discuss the angular correlation $\rho(\theta)$ Bertsch and Esbensen (1991); Hagino and Sagawa (2005b); Papadimitriou et al. (2011); Wang et al. (2017), which is defined as the probability to detect the two valence protons with an opening angle $\theta$.

The GCC prediction for the g.s. of 12O is shown Fig. 3. It is interesting to compare this result with $\rho(\theta)$ for 6Be Wang et al. (2017), which can be associated with a direct $2p$ decay Pfützner et al. (2012); Egorova et al. (2012); Wang et al. (2017). In both cases, a diproton-like structure corresponding to a peak at small opening angles is very pronounced. However, in 12O, the $2p$ angular correlation shows a rather weak angular dependence at large opening angles, and there is no pronounced minimum around 90∘. Moreover, the two valence protons are calculated to form different T-type Jacobi-coordinate configurations in these two nuclei. Namely, in the case of 12O the dominant $(S,\ell_{x},\ell_{y})$ configurations are 65% (0, 0, 0) and 20% (1, 1, 1), while they are 83% and 12%, respectively, for 6Be. This indicates that, besides diproton decay, there is another $2p$ decay mode – “democratic” decay – in the g.s. of 12O, in which two emitted protons are uncorrelated and may decay sequentially.

An important problem in the description of $2p$ emitters is the evolution of $2p$ correlations during the decay process. One way to look at this evolution is to calculate the $2p$ flux \mbox{\boldmathj}={\rm Im}({\Psi}^{\dagger}\nabla\Psi)\hbar/m, which shows how the two valence protons evolve within a given state $\Psi$. In our framework, the $2p$ flux can be computed readily as the wave function $\Psi$ is expressed in the Berggren basis. Results for 6Be and 12O are shown in Fig. 4. For both nuclei, the density distribution shows two maxima associated with diproton and cigarlike configurations.

In the case of 6Be shown in Fig. 4(a), the competition between diproton and cigarlike configurations exists inside the inner turning point of the Coulomb-plus-centrifugal barrier associated with the core-proton potential. (To estimate the centrifugal potential, we took the angular momentum of the dominant channel.) Near the origin, the dominant diproton configuration tends to evolve toward the cigarlike configuration because of the repulsive Coulomb interaction and the Pauli principle. On the other hand, near the surface the direction of the flux is from the cigarlike maximum toward the diproton one in order to tunnel through the barrier. Moreover at the peak of the diproton configuration which is located near the barrier, the direction of the flux is almost aligned with the core-$2p$ axis, indicating a clear diproton-like decay. Beyond the potential barrier, the two emitted protons tend to gradually separate due to the repulsive Coulomb interaction. The behavior of the two protons below the barrier can be understood by the influence of pairing which favors low angular momentum amplitudes; hence, it effectively lowers the centrifugal barrier and increases the probability for the two protons to decay by tunnelling Wang et al. (2017); Grigorenko et al. (2009); Oishi et al. (2014, 2017).

The case of 12O shown in Fig. 4(b) nicely illustrates the competition between direct and “democratic” $2p$ decay. Indeed, a significant part of the flux from the diproton configuration toward the cigarlike configuration persists up to the potential barrier and beyond. This indicates that some fraction of the decay is “democratic” despite the cigarlike configuration being far less dominant in 12O than in 6Be. One could thus expect pairing to play a lesser role in the decay process of 12O.

III.2 11O and its mirror partner 11Li

III.2.1 Spectra and angular correlations

In this section, we discuss the mirror pair of the proton-unbound 11O and the Borromean neutron halo 11Li. Selected GCC results on 11O can be found in Ref. Webb et al. (2019a). In order to benchmark the GCC model, we compare our calculations for 11Li with those using the core-neutron potential of Ref. Hagino and Sagawa (2005a) (GCC′ in Fig. 1). The g.s. energy, which is close to the experimental energy, and the angular correlation shown in Fig. 5 are both similar to the results of Ref. Hagino and Sagawa (2005a). The dineutron peak predicted in GCC′ is slightly broader than that predicted in Ref. Hagino and Sagawa (2005a) where a contact interaction between the valence neutrons was used.

Figure 5 also shows the angular correlations for the second 3/2- and first 5/2+ excited states. There are conspicuous differences between these correlations and those of the ground state. Neither of the excited states in 11Li has the separate dinucleon peak identifiable in the ground state. The second 3/2- in 11O displays a conspicuous large angle correlations often referred to as “cigar”-like. Moreover, there are two peaks in the 3/2${}^{-}_{2}$ state of 11O, while only one broad peak in its isobaric analog.

We point out that our calculated 5/2${}_{1}^{+}$ state of 11Li, with $E_{x}$ = 0.906 MeV and $\Gamma_{2p}$ = 0.258 MeV, is consistent with, and a candidate for, the lowest observed excitation in 11Li Kanungo et al. (2015); Tanaka et al. (2017).

In Fig. 1(c) we compare GCC and GCC’ results for 10N and 11O. The results are in a qualitative, if not quantitative, agreement. The measured energies and decay widths of the lowest states in 10N are well reproduced, and both models predict multiple states for 11O in the energy region where a broad structure was observed Webb et al. (2019a). As discussed in Ref. Webb et al. (2019a) using the the decay-width analysis, the observed structure in 11O almost certainly contains multiple components.

The calculated g.s. energy of 11O is 3.173 MeV ($\Gamma_{2p}$ = 0.861 MeV) and 2.613 MeV ($\Gamma_{2p}$ = 1.328 MeV) in the GCC and GCC′ variants, respectively. Both values are consistent with the estimated $2p$ decay energy $Q_{2p}$ = 3.21(84) MeV based on the extrapolation of the quadratic isobaric multiplet mass equation fit to the three neutron-rich members of the $A=11$ sextet Charity et al. (2012).

III.2.2 Threshold resonance in 10N

The fact that 10N and 11O are, respectively, the mirror nuclei of 10Li and 11Li offers the opportunity to revisit the question about the role that the antibound state in 10Li plays in the $2n$ halo structure in 11Li Thompson and Zhukov (1994); Betan et al. (2004); Michel et al. (2006); Simon et al. (2007); Aksyutina et al. (2008), but in the context of the proton-rich nuclei 10N and 11O. Hereafter, we investigate how the structure of 10N affects the 3/2${}^{-}_{1}$ g.s. of 11O.

In Ref. Webb et al. (2019a), we analyzed the $2p$ partial decay widths of the 3/2${}^{-}_{1}$ ground state of 11O as a function of $Q_{2p}$, which is controlled by the depth $V_{0}$ of the core-proton potential. It has been shown that below the Coulomb barrier the calculated decay width increases rapidly with $Q_{2p}$, as expected. This is accompanied by a rapid change of the dominant configuration with a discontinuity in the 3/2${}^{-}_{1}$ state trajectory as $Q_{2p}$ changes from 3.6 to 4.1 MeV. At energies above the barrier, the wave function has a small amplitude inside the nuclear volume. For example, when $Q_{2p}>3.5$ MeV, the 3/2${}^{-}_{1}$ solution has less than 20% of the total wave function inside a 10 fm radius. As $Q_{2p}$ increases further, this GCC solution cannot be traced anymore as the computation becomes numerically unstable.

In order to understand the role of the continuum in the g.s. of 11O, the shell model amplitudes $c(k)$ associated with the $s$ ($\ell=0$) partial wave in the 3/2${}^{-}_{1}$ state were extracted. As shown in the Fig. 6, continuum states have a large $s$-wave amplitude when $Q_{2p}$ approaches the Coulomb barrier at $k=0.393$ fm*-1*, which indicates strong continuum couplings at this $2p$ decay energy. This behavior is reminiscent of the situation in the mirror nucleus 11Li Michel et al. (2006), wherein an antibound state in the subsystem 10Li, viewed as $n$+9Li, is important 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).

The sharp change in the shell model amplitudes around the $2p$ threshold suggests that the system reorganizes itself as a consequence of the channel opening. To confirm this idea, knowing that the halo structure of 11Li is strongly affected by the antibound state in 10Li, we probe the link between the near-threshold resonant poles in 10N and properties of 11O. To this end, we follow the trajectory of the antibound state of 10Li in the complex-$k$ plane by gradually increasing the Coulomb interaction by changing the core charge $-Z_{c}e$ from zero ($n+^{9}$Li) to the full $p+^{9}$C value at $Z_{c}=6$. The results are shown in Fig. 7. At $Q_{2p}=4.13$ MeV, the antibound state of 10Li at $E=-1.02$ MeV is predicted by our model. With increasing $Z_{c}$, this pole goes through the region of subthreshold resonances defined by Re$(E)<0$ and $\Gamma>0$ and located below the $-45^{\circ}$ line in the momentum plane Kok (1980); Sofianos et al. (1997); Mukhamedzhanov et al. (2010), and eventually becomes a threshold resonant state in 10N at $Z_{c}=6$. This is not surprising as antibound states do not exists in the presence of the Coulomb interaction Efros and Bang (1999); Lovas et al. (2002); Csótó (2002). It is worth noting that our model also predicts the second subthreshold resonance, which slightly moves down in the complex-$k$ plane with $Z_{c}$.

As the WS potential becomes deeper ($Q_{2p}=2.09$ MeV) the antibound state in 10Li becomes a marginally bound halo state, which – with increasing $Z_{c}$ – becomes a decaying threshold resonant pole. (This is reminiscent of what happens in the two-nucleon system Kok (1980).) The examples shown in Fig. 7 suggest that the character of the isobaric analog of the antibound state of 10Li in 10N strongly depends on the strength of the core-nucleon interaction, or, alternatively, $Q_{2p}$.

As illustrated in Fig. 8, with decreasing $|V_{0}|$, the broad threshold resonant state in 10N is moving towards the unobservable region of subthreshold resonances with distinctively different asymptotic behavior. Since this state contributes the the g.s. wave function of 11O, it is difficult to handle its extended wave function numerically when solving for the 3/2${}^{-}_{1}$ state, which can result in a discontinuity. At the experimental value $Q_{2p}=4.13$ MeV ($V_{0}=-52.17$ MeV), the broad $s$-wave threshold resonant state in 10N is located at $k=0.252-0.213i$ fm*-1* ($E=0.38$ MeV, $\Gamma_{p}=4.45$ MeV), i.e., very close to the $-45^{\circ}$ line in the complex-$k$ plane. As $|V_{0}|$ decreases further, a second branch of the 3/2${}^{-}_{1}$ solution appears at higher $Q_{2p}$ values. This solution corresponds to a different configuration but it follows the first branch (Webb et al., 2019a). For the 3/2${}^{-}_{2}$ state, the trend is the opposite: the ($s_{1/2})^{2}$ amplitude in this state decreases rapidly with $Q_{2p}$ and this solution eventually can be associated with an almost a pure ($p_{1/2})^{2}$ configuration.

Using the available experimental information about 11O Webb et al. (2019a), the GCC calculation of the energy spectrum of 11O shown in Table 1 yields a g.s. value of $Q_{2p}=4.158$ MeV and a $2p$ decay width $\Gamma_{2p}=1.296$ MeV. The decay widths of the excited states (5/2${}^{+}_{1}$ and 3/2${}^{-}_{2}$) are all enhanced by about 50% as compared to the original energy spectra in Fig. 1 and are close in energy. At this value of $Q_{2p}$, the configuration of 11O is predicted to be fairly similar to that of 11Li. However, we want to emphasize the great sensitivity of the calculated configuration of 11O on the $2p$ decay energy. The differences between the structures of 11O and 11Li are clearly seen in different wave functions and angular correlations. For instance, as seen in Fig. 5, the diproton peak in the 3/2${}^{-}_{1}$ state of 11O, is very pronounced and most of the contributions to the angular correlation are coming from the pairing part ($S=0$). In 11Li, on the other hand, the $S=1$ component of the 3/2${}^{-}_{1}$ state is much larger.

IV Summary

The deformed core+nucleon+nucleon Gamow coupled-channel (GCC) approach has been used to describe the spectra and $2p$ emission of 11,12O. The model reproduces experimental low-lying states of 12O and its mirror system 12Be. The dynamics of the 2$p$ emission has been studied by analyzing the $2p$ flux in the ground states of 6Be and 12O. We conclude that in the case of 6Be the $2p$ emission has a diproton character while in 12O there is a competition between diproton and “democratic” decays.

For 11O, multiple excited states are predicted within the $Q_{2p}$ energy range from 3 MeV to 6 MeV. Moreover, we found the $3/2^{-}_{1}$ g.s. is strongly influenced by the existence of a broad threshold resonant state in 10N, which can be viewed as the isobaric analog of the antibound state in 10Li in the presence of the Coulomb potential.

According to our calculations, the energy spectra, shell-model wave-function amplitudes, density distributions, and angular correlations show significant differences between the unbound 11,12O and their mirror partners. In the case of the 11O and 11Li mirror pair, the Thomas-Ehrman effect is moderate for the CGG Hamiltonian optimized to experiment, but the results are highly sensitive to the $Q_{2p}$ energy assumed.

The future enhancements of the GCC model will pertain to the reliable description of $2p$ correlations in the momentum representation. This will allow a direct comparison between experimental angular distributions and theoretical $2p$ wave functions, and will help further constraining the effective Hamiltonian used.

Acknowledgements.

Discussions with Furong Xu are acknowledged. We appreciate helpful comments from Kévin Fossez. 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-SC0013365 (Michigan State University), DE-SC0018083 (NUCLEI SciDAC-4 collaboration), DE-SC0009971 (CUSTIPEN: China-U.S. Theory Institute for Physics with Exotic Nuclei), and DE-FG02-87ER-40316.