One-neutron halo structure of $^{29}$Ne

We have applied the Gamow shell model to calculate nuclear observables of $^{26-31}$Ne isotopes pertaining to one-neutron halo structure, these nuclei being situated close to neutron drip-line. As both many-body correlations and continuum coupling are taken into account in that approach, halo structure can be analyzed properly. Our calculations provide good descriptions of $^{26-31}$Ne, where asymptotic behavior is crucial for that matter. One-body density, neutron root-mean-square radii of $^{26-31}$Ne, and one-neutron overlap functions of $^{29,31}$Ne have been calculated as well. Our results support the presence of a one-neutron \textit{p}-wave halo in $^{31}$Ne, already pointed out experimentally. A similar situation also occurs in the ground state of $^{29}$Ne, which is mainly a \textit{p}-wave valence neutron coupled to the inner $^{28}$Ne \textit{core}. The $3/2^+$ excited state of $^{29}$Ne, which is dominated by a \textit{d}-wave valence neutron, has also been considered. A larger radius and more extended wave function occur for the ground state of $^{29}$Ne when compared to its $3/2^+$ first excited state. The present results suggest that $^{29}$Ne is a good candidate for one-neutron \textit{p}-wave halo in the medium-mass region.

and 22 C [16] as well as the one-neutron halo nuclei 11 Be [17] and 15,19 C [18,19]. Only a few halo nuclei are known in the medium mass region, such as the one-neutron halo present in 31 Ne [20,21] and 37 Mg [22], and the two-neutron halo in 29 F [23]. The existence of halo structure implies that significant modifications in the shell structure of these nuclei are generated. In particular, they involve mixing in their ground states induced by intruder states, such as s or p partial waves [8]. Intruder state mixing can lead to considerable deformations in these states. Coupling to higher partial waves via configuration mixing can also occur. Hence, continuum coupling and many-body correlations must be treated properly for a proper description of halo nuclei [8]. In Gamow shell model (GSM) [7,[24][25][26][27][28], both continuum coupling and many-body correlations are included, so that it is an appropriate tool to precisely study the exotic properties of drip-line nuclei, in particular halo structures [29]. In fact, GSM [7,24,25] has become a powerful and predictive tool to determine the nuclear structure and reaction observables of drip-line nuclei [30,31].
Neon isotopes have been synthesized up to the neutron dripline, as it is reached via the drip-line nucleus 34 Ne [20,32]. Its rich structure has also been established experimentally [21,[33][34][35]. The neon chain forms an interesting ground for theoretical studies, as they can provide information to understand the proton-neutron and neutron-neutron interactions, as well as many-body correlations at drip-lines. Moreover, 31 Ne is a oneneutron halo [21,36], as the valence neutron of 31 Ne is found to occupy p-waves, instead of f -waves, the latter situation being expected from standard shell model [37]. As it has a small oneneutron separation energy, of about 200 keV, a one-neutron halo can form due to the coupling of a p-wave valence neutron cou-pled with the deformed 30 Ne inner core. Added to that, the 3/2 − ground state of 29 Ne exhibits large p-wave intruder configuration components [33,34] in conjunction to the slight deformation found in the ground state of 28 Ne [33,35]. Conversely, the first 3/2 + excited state of 29 Ne is dominated by localized configurations. The one-neutron separation energy (S n ) of 29 Ne is about 900 keV [33], which is larger than the one-neutron separation energy S n of 31 Ne, which is 200 keV [21,32]. However, S n in 29 Ne is comparable to the one-neutron s-wave halo nuclei 15,19 C [32], which are about 1200 and 700 keV, respectively. Unfortunately, contrary to 31 Ne, there is scarce experimental and theoretical information available on the ground state of 29 Ne, which is likely a one-neutron halo. The similar structures of 29 Ne and 31 Ne can surely give rise to interesting phenomena and shed light on the suspected one-neutron halo properties of 29 Ne.
In GSM, many-body correlations are included via configuration mixing and continuum coupling is present at basis level [7]. Therefore, GSM is suitable for the description of exotic properties in drip-line nuclei. It is the object of the present paper to study neon isotopes situated close to neutron drip-line with GSM, in order to investigate the halo properties of 31 Ne and 29 Ne. The present paper is written as follows. The basic framework of GSM is firstly introduced briefly. Then, we present the model space and many-body Hamiltonian of GSM used in the present work. In order to ponder about the halo structures of 31 Ne and 29 Ne, we will show the one-body densities, neutron root-mean-square (rms) radii, and single-particle (s.p.) overlap functions of neon isotopes obtained from GSM calculations. Conclusion will be made afterwards.
Model. -The fundamental theoretical construction entering GSM is the one-body Berggren basis. The Berggren basis was initially proposed by T. Berggren in Ref. [38]. It consists of bound, resonance and scattering s.p. states, generated by a finite-range potential, such as Woods-Saxon (WS) potential. The completeness of the basis [38] for a given partial wave of quantum number j reads : n |n j n j| + L + |k j k j| dk = 1 (1) where |n j runs over resonant (bound or resonance) states, and L + is a complex contour in the complex-momentum space encompassing resonance states. The L + integral contains the continuous part of the Berggren basis, with |k j running over the scattering states belonging to the L + contour. In numerical GSM calculations, the L + integral is efficiently discretized with the Gauss-Legendre quadrature, as convergence is obtained with 20-30 points [7,26,27]. From the Berggren basis of Eq. (1), a many-body basis of Slater determinants can be generated, so that internucleon correlations are described via configuration mixing in a shell-model picture [7, 24-26, 26, 27, 39]. Hence, GSM is a suitable approach to study many-body weakly bound and unbound states. GSM is usually performed in the picture of a core plus valence particles. In order to properly account for center-of-mass degrees of freedom, one uses the cluster-orbital shell model framework (COSM) [40]. Nucleon COSM coordinates are defined with respect to the center-of-mass of the core, so that one works in a translationally invariant frame, thus where center-ofmass excitations cannot develop. The GSM Hamiltonian using COSM reads : where N val is the number of valence nucleons, µ i is the reduced mass of the i-th nucleon with respect to the core, and M c is the mass of the core. The one-body potentialÛ i is represented by a WS potential mimicking the inert core. V i j is the two-body residual interaction, which is modeled for our present work by a pionless effective field theory (EFT) interaction [41][42][43]. The part proportional top i ·p j is an additional two-body kinetic term, which takes into account the recoil of the active nucleons relative to the chosen core in the COSM framework. Concerning the used pionless EFT interaction [41][42][43], only two-body contact terms up to next-to-next leading-order are considered in the present calculations. EFT parameters are optimized to reproduce the energies of the low-lying states of selected nuclei. The harmonic oscillator (HO) basis used for the representation of the EFT interaction is limited to a few shells. Thus, the highmomentum components of the initial EFT interaction are automatically suppressed. This regularization approach has been recently utilized in Refs. [44][45][46][47], and in particular in GSM with the calculation of neutron-rich oxygen isotopes [28]. Table 1: Optimized parameters of the EFT interaction at leading order (LO) and next-to-leading order (NLO). They are given in natural units. The C 0,1 S , C 0,1 T , C 1...7 notations are taken from Refs. [41][42][43] . The parameters at leading order (C 0,1 S , C 0,1 T ) explicitly depend on the isospin T = 0, 1 of the two nucleons.
The closed-shell nucleus 24 O is used as the inert core in this work. As valence protons are well bound in neutron-rich neon isotopes, it is sufficient to consider the harmonic oscillator states 0d 5/2 and 1s 1/2 to generate the proton part of the model space. For valence neutrons, the d 3/2 , f 7/2 , p 3/2 and p 1/2 partial waves are represented by the Berggren basis, as they are most crucial to generate a proper coupling to the continuum. The parameters of the one-body WS potential and two-body EFT force in the GSM Hamiltonian are optimized to reproduce the low-lying states of 25,26 O, 25,26 F and 26−29 Ne. More precisely, all the ground states and first excited states of these isotopes were fitted, except for the first excited state of 25 O. The third and fourth excited states of 26 F have also been added to the fit. The fixed parameters of the WS core potential are the diffuseness d = 0.65 fm, the radius R 0 = 3.663 fm and the spin-orbit coupling V s = 7.5 MeV. The central depth V 0 differs according to partial waves : it is equal to 65.81 and 62.21 MeV for proton s 1/2 and d 5/2 partial waves, whereas, for the neutron part, it is 40 MeV for = 0, 2, 47 MeV for = 1 and 38 MeV for = 3 partial waves. The resulting fit of the parameters of the pionless EFT interaction is shown in Tab. 1. The EFT interaction is separated into two parts: the leading-order part (C 0,1 S and C 0,1 T ) and the next-to-leading order (C 1...7 ) part (see Refs. [41][42][43] for notation and definition of associated operators). The C S and C T constants depend on the isospin T = 0, 1 of the two nucleons. As C S and C T reduce to a single constant in our framework, we only fitted C S and arbitrarily set C T = 0. The obtained Hamiltonian provides the neutron separation energy of 29 Ne to about 840 keV, which is consistent with its experimental value of 900 keV [33]. We also performed GSM calculations for 30,31 Ne using the optimized Hamiltonian. S n in 31 Ne is about 350 keV, which is also close to its experiment value of 200 keV. Consequently, as we obtained a good agreement of the one-neutron separation energies S n in 29,31 Ne with experimental data, we can proceed to the quantitative investigation of the halo properties of 29,31 Ne.
Results. -The one-neutron p-wave halo nucleus 31 Ne has been experimentally studied using 1n-removal reactions [21]. It was concluded that the one-neutron halo 31 Ne isotope is deformation-driven, of p-wave character, and exhibits a small S n of 150 +160 −100 keV [21]. Recent experiments [33,34] showed that the ground state of 29 Ne is dominated by p-wave intruder configurations with J π = 3/2 − , where S n is about 900 keV. Whether 29 Ne exhibits one-neutron halo properties remains, however, to be determined. Moreover, the inner core 28 Ne in 29 Ne is not so much deformed as the inner core 30 Ne of the oneneutron halo nucleus 31 Ne [33,35]. In fact, due to their subtle interplay of continuum coupling and deformation, the 29,31 Ne isotopes offer unique prototype systems to study the mechanism of shell evolution and halo formation in the medium-mass region.
In one-neutron halos, the neutron density distributions are extremely extended when compared to those generated by their associated inner cores in the asymptotic region. To analyze the one-neutron halo structures in 29,31 Ne isotopes, we calculated the one-body densities of the ground states of the 29,31 Ne isotopes and those of the corresponding inner cores 28,30 Ne with GSM. The obtained results are shown in Fig. 1. We can clearly therein see that the one-body density of the ground state of 31 Ne decreases slowly when compared with that of its inner core 30 Ne in the asymptotic region. Our above results can be seen to be similar to the one-body densities of the one-neutron s-wave halo 15 C [18] and one-neutron p-wave halo 37 Mg [9]. The calculated one-body density of the 31 Ne is also consistent with experimental data and supports the one-neutron p-wave halo character of the 31 Ne nucleus [21]. A similar situation occurs in 29 Ne, where the one-body neutron density also decreases slowly in the asymptotic region when compared to that of its inner core 28 Ne. Therefore, the similar structure of the one-body densities of 29 Ne and 31 Ne suggests that 29 Ne is a good candidate for one-neutron halo in the medium-mass region.
To further analyze the halo structures of 29,31 Ne, and to provide deeper comparisons between the well-known one-neutron halo nucleus 31 Ne with the halo candidate 29 Ne, we calculated the neutron rms radii of the ground states of 26−31 Ne isotopes  with GSM. The results are presented in Fig. 2. We can see that the calculated neutron rms radii of 29 Ne and 31 Ne do not follow the line formed by the neutron rms radii of other neon isotopes, 29,31 Ne rms radii being situated 0.05 to 0.1 fm above this line. The one-neutron halo of 31 Ne is thus clearly shown by comparing its neutron rms radius to that of its inner core 30 Ne. Our results are in accordance with those of Ref. [21]. A similar situation also occurs in the one-neutron halo 19 C, as the experimental rms radius of 19 C is larger than that of its inner core 18 C [19]. To assess the effect of nuclear structure on neon rms radii, the neutron rms radius of the first 3/2 + excited state of 29 Ne has been calculated for comparison. Indeed, the latter is mainly dominated by rather localized shell model configurations, where d-waves are mostly occupied [37]. GSM calculations provide a neutron rms radius of the 29 Ne ground state (3/2 − ) which is larger than that of its first 3/2 + excited state. The large radius difference between the 3/2 − ground state and 3/2 + excited state of 29 Ne thus provides an additional argument in favor of the presence of a one-neutron p-wave halo in the 29 Ne ground state.
GSM results for neutron rms radii of neon isotopes, increasing from 26 Ne to 30 Ne, also suggest the disappearance of the shell closure N = 20 in the neon chain. GSM results follow the conclusions derived from mass and spectra data, as the trend of neutron rms radii of neon isotopes is akin to that in magnesium isotopes of the island of inversion [48]. Furthermore, the obtained neutron rms radii of 29,31 Ne are also consistent with experimental interaction cross sections, as they give significantly greater values in 29,31 Ne than in neighboring nuclei [49]. However, our calculation of the neutron rms radius of 27 Ne is not consistent with the slight enhancement of interaction cross section seen in 27 Ne, which may be due to the lack of configuration-mixing in the present calculations, using a 24 O core. In one-neutron halos, spectroscopic factors of the valence neutron above the inner core are typically large, with a value comparable to those associated to s.p. states close to doublemagic nuclei. As continuum coupling plays an essential role in the calculations of spectroscopic factors in weakly bound and unbound nuclei, GSM is a proper model for their evaluation [50]. In fact, despite their non-observable nature, the study of spectroscopic factors in halo nuclei is crucial to understand the exotic properties of nuclear systems close to drip-lines. Onenucleon radial overlap functions can also provide important information about one-neutron halo nuclei, due to the fact that they resemble a valence nucleon wave function weakly coupled to an inner core [51].
In the GSM framework, the one-nucleon radial overlap function [50] is defined as : where a † (B) is a creation operator associated with the s.p. basis state |u B . The tilde symbol above the bra-vector reminds that complex conjugation is absent from matrix elements when using the Berggren basis, which can be rigorously used only in rigged Hilbert spaces [26]. Spectroscopic factors can be computed via the relation : Spectroscopic factors are complex in GSM. The interpretation of complex observables is that their real part corresponds to the statistical average of all made measurements, while their imaginary part corresponds to their statistical uncertainty (see Ref. [26] and references therein). In our calculations, the imaginary part of spectroscopic factors is very small, so that they can be neglected in our present analysis. As the GSM calculations of one-nucleon overlap functions and spectroscopic factors are done by summing over all discrete Gamow states and discretized scattering states along the L + contour [7], they are independent of the used s.p. basis [50]. This is not the case in HO shell model calculations, performed in a limited model space where usually only one state of a given partial wave is present [52]. The one-neutron overlap functions of the ground states of 29,31 Ne and the first 3/2 + excited state of 29 Ne corresponding to the ground states of 28,30 Ne are calculated. Results are shown in Fig. 3. One-neutron halo structure can be clearly seen from the extended one-neutron overlap function in the asymptotic region. As the ground states of 29 Ne and 31 Ne have many-body quantum numbers J π = 3/2 − , they mainly consist of an inner core coupled to a p-wave valence neutron. The first 3/2 + excited state of 29 Ne, on the contrary, is dominated by the inner core coupled to a d-wave valence neutron. The comparison of the one-neutron overlap function of 29 Ne with the well-known halo 31 Ne is essential for the predictions of the halo structure of 29 Ne. We can see from Fig. 3 that the one-neutron overlap function associated to the neutron p 3/2 partial wave in the ground state of 29 Ne is similar to that of 31 Ne. Furthermore, the overlap function of the ground state of 29 Ne shows a slower decrease than that of the d 3/2 one-neutron overlap function of the first 3/2 + excited state of 29 Ne. This situation arises as p 3/2 components are more extended in the asymptotic region than d 3/2 components because of their different centrifugal parts. Hence, GSM one-neutron overlap functions also support the p-wave one-neutron halo character of the ground state of 29 Ne.
We calculated p 3/2 spectroscopic factors of 29,31 Ne from the obtained one-neutron overlap functions. We obtained the values 0.540 and 0.392 for 29 Ne and 31 Ne, respectively. Both these results are close to experimental data, which are equal to 0.54 (9) [33] and 0.32 +21 −17 [21], respectively. The calculated neutron p 3/2 spectroscopic factor in 29 Ne is larger than that of 31 Ne, which is consistent with experiment data. Nuclear deformation can be described via a configuration-mixing shell-model picture using a large valence space. Stronger d 3/2 and p f cross-shell configuration mixing in the 30 Ne ground state is obtained in our GSM calculation compared to the 28 Ne ground state. Results suggest that deformation in 30 Ne is larger than that in 28 Ne. A similar situation is obtained in HO shell model calculations [33]. Furthermore, a large deformation arising from a strong configuration mixing in the inner core of one-neutron halo nucleus typically provides a smaller spectroscopic factor value. The experimental and theoretical spectroscopic factors of 29,31 Ne indicate that the deformation of the core 28 Ne in the one-neutron halo nucleus 29 Ne is smaller than that of the core 30 Ne in the one-neutron halo nucleus 31 Ne.
We checked that the obtained results do not qualitatively change with other parametrizations of the Hamiltonian. For this, one used the Hamiltonian of Ref. [10], where the one-body WS part was slightly refitted in order for the separation energies of 29,31 Ne to be reproduced. The general trend observed in densities, rms radii and overlap functions (see Figs. 1,2, and 3) is indeed also present with the Hamiltonian of Ref. [10], these values being quantitatively different from those illustrated in this paper by about 10% at most. Consequently, the one-neutron halo structure found in 29 Ne is very likely to occur experimentally, as it is reproduced by two different Hamiltonians, where parameters are fitted only on energies and where densities, rms radii and overlap functions are predicted. Such a situation had already been encountered in Ref. [10], where part of the authors investigated the possibility of a two-neutron halo of 31 F with GSM.
Summary. -Drip-line nuclei exhibit unique phenomena, such as halo structure. The multiconfigurational approach GSM, in which both many-body correlations and continuum coupling are included, has been used to study the one-neutron halo character of 29 Ne and 31 Ne. The one-neutron separation energies of 29,31 Ne, which are essential in the formation of their halos, are well reproduced by using a Hamiltonian whose parameters are fitted from nearby nuclei. One-body densities, neutron rms radii, and one-neutron overlap functions of neutron-rich neon isotopes have also been calculated. The wellknown one-neutron halo 31 Ne happens to be well described in our GSM calculations. By comparing the calculated one-body density, neutron rms radius, and one-neutron overlap function of 29 Ne, to those of the well known one-neutron halo 31 Ne, we could establish that 29 Ne and 31 Ne exhibit halo-dependent ob-servables of similar value and shapes. The neutron rms radius and one-neutron overlap functions of the first 3/2 + excited state in 29 Ne have been calculated for comparison. Their qualitative difference with those of the ground state of 29 Ne also supports its one-neutron halo character. As an additional test for that matter, spectroscopic factors involving the weakly bound valence neutron in 29,31 Ne have been computed using the obtained one-neutron overlap function. Results have been found to be consistent with experimental values. Consequently, our calculations suggest that the ground state of 29 Ne exhibits a oneneutron p-wave halo, and, hence, that it is a good candidate for one-neutron halo nucleus in the medium-mass region.