Shrunk halo and quenched shell gap at $N=16$ in $^{22}$C: Inversion of $sd$ states and deformation effects

We investigate the ground state properties of $^{22}$C by using a deformed relativistic Hartree-Bogoliubov model in continuum and explore the interplays among the formation of a halo, deformation effects, the inversion of $sd$ states, the shell evolution, and changes of nuclear magicities. It is revealed that there is an inversion between the two spherical orbitals $2s_{1/2}$ and $1d_{5/2}$ in $^{22}$C compared with the conventional single particle shell structure in stable nuclei. This inversion, together with deformation effects, results in a shrunk halo and a quenched shell gap at $N=16$. It is predicted that the core of $^{22}$C is oblate but the halo is prolate. Therefore several exotic nuclear phenomena, including the halo, the shape decoupling effects, the inversion of $sd$ states, and the evolution of shell structure which results in (dis)appearance of magic numbers, coexist in one single nucleus $^{22}\mathrm{C}$.


Introduction
The study of exotic nuclear structure is at the forefront of research in modern nuclear physics [1]. Among many others, the most striking exotic nuclear phenomenon is the nuclear halo which was first observed in 11 Li [2]. Halo nuclei are weakly bound and well associated with pairing correlations and the contribution of the continuum above the threshold of particle emission [3,4,5,6,7,8,9,10,11,12]. The formation of a nuclear halo is closely connected with the evolution of the shell structure and changes of nuclear magicities around drip-lines [13,14,15,16].
Most known nuclei are deformed with shapes deviating from a sphere. When the deformation is involved in, even more exotic phenomena are expected [17]. The shape decoupling phenomenon, i.e., the core and the halo having different shapes, has been predicted in deformed nuclei close to the neutron drip-line [18,19]. For example, in 42,44 Mg, the core and the halo are predicted to be prolate and oblate, respectively. Such predictions were made by using a deformed relativistic Hartree-Bogoliubov model in continuum (DRHBc model) [18,19,20] which describes self-consistently the large spatial extension, the contribution of the continuum due to pairing correlations, and deformation effects in deformed nuclei with halos. Later similar shape decoupling effects were also revealed by using nonrelativistic Skyrme Hartree-Fock-Bogoliubov models for axially deformed nuclei in coordinate space [21,22,23,24] or in a Gaussian basis [25,26].
As the heaviest Borromean nucleus with a halo observed so far, 22 C is of particular interest because of not only possible new magicities but also uncertainties and puzzles in the separation energy, the matter radius, and the halo configuration. If the Z = 6 magic number evidenced in neutron-rich C isotopes [27] persists in it and the shell gap at N = 16 is large enough, 22 C could be a new doubly magic nucleus [28]. The empirical value of the two-neutron separation energy S 2n is 420(940) keV in AME2003 [29] and 110(60) keV in AME2012 [30,31,32]. In 2012, S 2n was determined to be −0.14 ± 0.46 MeV from direct mass measurements [33]. According to the recent AME2016, S 2n = 35 (20) keV [34,35,36]. The matter radius of 22 C deduced from interaction cross sections measured in two experiments differ very much: r m = 5.4 ± 0.9 fm in 2010 [37] and r m = 3.44 ± 0.08 fm in 2016 [38]. Recently, the determination of 22 C radius with interaction cross sections was re-examined by using the Glauber model and r m = 3.38 ± 0.10 fm was extracted [39]. In almost all investigations on 22 C [40,41,42,43,37,44,45,46,47,48,49,50,51,52,53,54,55,56,38,57,58,59,60,61,39], the two valence neutrons are assumed to occupy mostly the second s orbital 2s 1/2 . There are strong interplays among S 2n , r m , and the halo configuration, see, e.g., Ref. [62] for a recent review. An apparent puzzle arises from these interplays: if the two valence neutrons occupy 2s 1/2 and S 2n is very small, say, from several tens keV to several hundreds keV, the radius of 22 C should be much larger than the recent experimental value.
In this work, we study 22 C with the DRHBc model. It is shown that the 2s 1/2 orbital is a bit lower than the 1d 5/2 orbital when the spherical symmetry is imposed, i.e., these two states are inverted compared with the conventional shell structure in stable nuclei. The near degeneracy of 2s 1/2 and 1d 5/2 would lead to a large shell gap at N = 16. However, the ground state of 22 C is deformed. The inversion of (2s 1/2 , 1d 5/2 ), together with deformation effects, results in a shrinkage in the halo and a quenched shell gap at N = 16 in 22 C, thus resolving the puzzles concerning the radius and halo configuration in this exotic nucleus. Furthermore, we predict that the core of 22 C is oblate but the halo is prolate, adding one more candidate of deformed halo nuclei with shape decoupling effects.

The DRHBc model
The details of the DRHBc model with nonlinear meson-nucleon couplings can be found in Refs. [18,19,20]. The DRHBc model with density-dependent couplings has been developed by Chen et al. [63]. Here we only present briefly the formalism for the convenience of the following discussions. In the DRHBc model, the RHB equation for nucleons [64] is solved in a Woods-Saxon (WS) basis [65] which can describe the large spatial extension of halo nuclei. In Eq. (1), h D is the Dirac Hamiltonian, λ is the chemical potential, and E k and (U k , V k ) T are the quasiparticle energy and wave function. The pairing potential reads, with a density dependent force of zero-range and the pairing tensor κ(r 1 , r 2 ) [66,67]. In the Dirac Hamiltonian [68,69,70,71,72,73,74,75,76] the scalar and vector potentials are expanded in terms of the Legendre polynomials, so are various densities in the DRHBc model. Note that for triaxially deformed nuclei, the expansion of potentials and densities should be made in terms of spherical harmonics [77].
Our calculations are carried out with the covariant density functional PK1 [78]. Since a zero-range interaction (3) is used in the pp channel, the pairing strength V 0 is connected with a truncation in the quasiparticle space. The Borromean feature of 22 C is used to fix the pairing parameters as: ρ sat = 0.152 fm −3 , V 0 = 355 MeV·fm 3 , and a cut-off energy E q.p. cut = 60 MeV in the quasi-particle space. These parameters result in S n = −28 keV for 21 C and S 2n = 0.43 MeV for 22 C.

Results and discussions
In Fig. 1, we display the density profiles of 22 C. The density distribution of the protons and neutrons are shown in the left and right parts of Fig. 1(a), respectively. It is clearly seen that the neutrons extend spatially much farther than the protons, hinting a neutron halo in 22 C. The calculated matter radius r m = 3.25 fm is significantly smaller than the experimental value 5.4 ± 0.9 fm given in 2010 [37] but close to the value 3.44 ± 0.08 fm obtained in 2016 [38] and 3.38 ± 0.10 fm extracted recently [39].
It should be mentioned that the empirical radius formula r m = 1.2A 1/3 fm gives 3.36 fm for A = 22 isobars [79]. This fact indicates that the halo in 22 C is not so pronounced if we adopt r m values from Refs. [38,39] or from our calculations. Having in mind that the two-neutron separation energy S 2n is quite small (≤ 0.5 MeV) as we have mentioned, such "small" r m values are quite puzzling if one accepts the assumption that the valence neutrons in 22 C occupy mostly the 2s 1/2 state. Next we address this issue by examining the halo configuration.
The augmented Lagrangian method [80] was implemented in the DRHBc model and deformation constraint calculations are carried out for 22 C. In  are in the continuum. These states contribute mostly to the halo and its deformation in 22 C as we will show later. From Fig. 2 one can find that 1/2 + 3 becomes more deeply bound with β 2 increasing from the ground state and joins 1d 5/2 with ε can ∼ −3.6 MeV at β 2 = 0. On the other hand, from the ground state to the spherical limit, 3/2 + 2 and 1/2 + 4 get closer in energy and finally merge as 1d 3/2 which is around 1 MeV above the threshold. The single neutron levels in the canonical basis in the spherical limit and at the ground state are also shown in Fig. 3.
It is interesting to see in Figs. 2 and 3 that, in the spherical limit, the 2s 1/2 state is lower than 1d 5/2 , i.e., these two states are inverted compared with the   conventional shell structure in stable nuclei. This inversion, together with the large spin-orbit splitting between the two d states, results in a noticeable shell gap at N = 16 when 22 C is constrained to be spherical. The inversion of (2s 1/2 , 1d 5/2 ) has been predicted in A/Z ∼ 3 nuclei [81] and the appearance of the N = 14 and N = 16 shell closures is closely related to the competition of 2s 1/2 and 1d 5/2 [81,82,83,84,85,28,86,87,88,16]. In Ref. [56], it is shown that by decreasing the parameter t 0 in the Skyrme interaction SIII, the 2s 1/2 orbital approaches 1d 5/2 and finally can be lower than the latter in 22 C.
It has been well accepted that the inversion of (2s 1/2 , 1d 5/2 ) results in the formation of the halo in 11 Li where the 2s 1/2 orbital is close to 1p 1/2 [89,5,90,91]. This inversion, however, plays an opposite role in 22 C: It hinders the halo formation when we stick to the spherical limit because the valence neutrons occupy a d-wave orbital. However, there are strong quadrupole correlations which drive 22 C to be well deformed with β 2 = −0.27. On the one hand, the deformation effects mix the sd orbitals, increase the neutron level densities around the Fermi surface, and destroy the N = 16 shell closure as is seen in Figs. 2 and 3. On the other hand, the mixture of the sd orbitals results in non-negligible 2s 1/2 components in 1/2 + 3 and 1/2 + 4 which are either weakly bound or in the continuum. The total amplitude of the 2s 1/2 component is about 25% in these two 1/2 + orbitals. Having in mind the degeneracy two, this means that about half of the valence neutrons is of the 2s 1/2 nature. Therefore the neutron halo in 22 C is shrunk compared with what it would be if the halo configuration is dominated by (2s 1/2 ) 2 .
In Figs 1/2 + 3 . The orbital 3/2 + 1 and those below it are deeply bound and contribute to the "core". The orbital 1/2 + 3 and those above it, the sum of occupation probabilities of which being 1.03, are weakly bound or in the continuum and form the "halo". In such a way we can decompose the neutron density into two parts. The density profiles of the neutron core and halo are presented in Figs. 1(b) and (c), respectively. It is clearly seen that the core of 22 C is oblate and the halo is prolate. This provides one more example of deformed nuclei with a shape decoupling besides 42 Mg and 44 Mg, both with a prolate core but an oblate halo [18,19].
In Fig. 4 the densities of the core and the halo of 22 C are decomposed into spherical (λ = 0), quadrupole (λ = 2), and hexadecapole (λ = 4) components [cf. Eq. (5)]. In Fig. 4(a), it can be found that the quadrupole component of the core is always negative, which corresponds to the oblate shape of 22 C. However, as seen in Fig. 4(b), although it is negative when r < 1.6 fm, the quadrupole component for the halo is mostly positive, which is consistent with what we have seen in Fig. 1(c), i.e., the halo of 22 C has a prolate deformation. From the slope of ε can as a function of β 2 around the ground state, it can be deduced that the wave function of the state 1/2 + 3 is prolate and that of 3/2 + 2 is oblate. Since it is dominated by 1/2 + 3 , the halo in 22 C is prolate.

Conclusions
In summary, to resolve the puzzles concerning the radius and configuration of valence neutrons in 22 C, the ground state properties of 22 C are studied by using a deformed relativistic Hartree-Bogoliubov model in continuum with the covariant density functional PK1. 22 C is predicted to be well deformed with an oblate shape. The neutrons extend spatially much farther than the protons. The calculated matter radius r m = 3.25 fm is fairly close to the two recent experimental values 3.44 ± 0.08 fm [38] and 3.38 ± 0.10 fm [39] but much smaller than the experimental value 5.4 ± 0.9 fm [37]. Deformation constraint calculations reveal that in the spherical limit the two orbitals 2s 1/2 and 1d 5/2 are inverted in 22 C compared with the conventional single particle level scheme in stable nuclei. This inversion hinders the halo formation if 22 C is constrained to be spherical. However, strong quadrupole correlations mix the sd orbitals. This mixture results in sizable 2s 1/2 components in valence neutron orbitals which are either weakly bound or in the continuum and leads to a shrunk halo in 22 C. The deformation effects also increase the neutron level densities around the Fermi surface and destroy the N = 16 shell closure. The neutron density is decomposed into the core and halo. It is found that the core of 22 C is oblate but the halo is prolate. Thus this nucleus becomes a new candidate of deformed halo nuclei with shape decoupling effects. Finally we mention that the present study was based on the effective interaction PK1 which is of meson-exchange and it will be interesting to make similar investigations with point-coupling interactions, e.g., PC-PK1 [92].