Prediction of two-neutron halos in the N = 28 isotones 40Mg and 39Na

Phys. Lett. B 853 (2024) 138694 Available online 7 May 2024 0370-2693/© 2024 The Author(s). Published by Elsevier B.V. Funded by SCOAP³. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Contents lists available at ScienceDirect Physics Letters B journal homepage: www.elsevier.com/locate/physletb Letter Prediction of two-neutron halos in the 𝑁=28isotones 40Mg and 39Na Jagjit Singh a,b,c,d, ,∗, J. Casal e,, W. Horiuchif,g,h,i,, N.R. Walet a,, W. Satuład, aDepartment of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK bDepartment of Physics, Akal University, Talwandi Sabo, Bathinda, Punjab 151302, India cResearch Centre for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan dInstitute of Theoretical Physics, Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warsaw, Poland eDepartamento de Física Atómica, Molecular y Nuclear, Facultad de Física, Universidad de Sevilla, Apartado 1065, E-41080 Sevilla, Spain fDepartment of Physics, Osaka Metropolitan University, Osaka 558-8585, Japan gNambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka Metropolitan University, Osaka 558-8585, Japan hRIKEN Nishina Center, Wako 351-0198, Japan iDepartment of Physics, Hokkaido University, Sapporo 060-0810, Japan A R T I C L E I N F O A B S T R A C T Editor: B. Balantekin The ground states of the nuclei 40Mg and 39Na are investigated using the hyperspherical formalism. Since they are located at the edge of the “big island of inversion”, we concentrate on whether we are likely to find twoneutron Borromean halos in these nuclei. A three-body model with effective 𝑛-𝑛and 38Mg+𝑛interactions is built for 40Mg based on the available data. We also give predictions for the low-lying spectrum of 38Na = 37Na + 𝑛and two-neutron separation energy of the 39Na nucleus. Depending on parameter choice, we report an increase in the matter radii in the range 0.1-0.5fm relative to those of the core nuclei. The results suggest a two-neutron halo structure in 40Mg for a subset of parameters, reinforcing the prediction of a Borromean halo nucleus. The calculations indicate that a two-neutron halo is even more likely for 39Na. As expected, the halo is linked to the disappearance of the shell gap in these nuclei due to the inversion of the 2𝑝3∕2 and 1𝑓7∕2 orbitals. We study the total cross section for scattering of these nuclei from a carbon target using a Glauber model and show that these provide a clear signal to assess the halo structure. 1. Introduction The development of novel radioactive ion beam facilities has made it possible to study the shell evolution across the nuclear chart, and, most interestingly, near the neutron drip line. Neutron-rich nuclides display exotic structural features such as dampening of shell gaps [1–3], shape coexistence [4–6]and formation of halos and deformed structure [7–13]. Above 𝑁=20, an explanation of the magic numbers requires a spinorbit force [14]. Thus, 𝑁=28 is the smallest magic number that is sensitive to the spin-orbit force. As shown in the lower panel of Fig. 1, for 20 ≤𝑁≤28 the 1𝑓7∕2 orbit is filled, while in the absence of spinorbit splitting the 1𝑓7∕2 and 2𝑝3∕2 orbits are almost degenerate. With modifications to the spin-orbit force, the spacing between these two orbits may change, and this can lead to an “inverted” order, where the 2𝑝3∕2 lies below the 1𝑓7∕2. The same result may also be caused by deformation [1,15]. Such islands of inversion (IOI) have been found in * Corresponding author at: Department of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK. E-mail address: [email protected] (J. Singh). isolated cases. It has also been shown that for 𝑁=20and 28 the IOIs merge in the neon, sodium, and magnesium isotopic chains, creating a “big island of inversion”, called the B-IOI [16,17]. The disappearance, or “melting”, of 𝑁=28 shell gap has been confirmed experimentally in neutron-rich Al, Si, S, and Ar isotopes [12,17–25]. For 𝑍=11, the lowest proton number in the B-IOI, the heaviest observed neutron-rich nucleus is 39Na [26,27], but the possibility of the existence of even heavier Na isotopes has not yet been excluded. Very few studies [27–29]examine the melting of 𝑁=28shell gap in Na and Mg isotones, among these only [27]delving into halo formation in 39Na using the deformed relativistic Hartree-Bogoliubov framework. The recent observation of the disappearance of the 𝑁=20shell gap at the low-𝑍side of the 𝑁=20chain (shown in the red circle in the upper panel of Fig. 1), has led to the identification of the 29Fsystem as the heaviest known two-neutron Borromean-halo nucleus [13,30–32]. Motivated by this observation, it is interesting to explore the low-𝑍side of https://doi.org/10.1016/j.physletb.2024.138694 Received 12 January 2024; Received in revised form 14 March 2024; Accepted 30 April 2024 Physics Letters B 853 (2024) 138694 2 J. Singh, J. Casal, W. Horiuchi et al. Fig. 1. (upper panel) A selected area of the nuclear Segrè chart for the 9 ≤ 𝑍≤14 isotopes. The green and orange blocks correspond to neutron numbers 𝑁=20 and 28, respectively. (lower panel) A sketch of the scenarios for the single-particle levels that lead to shell inversion. On the left, we show the shell gaps for stable nuclei (normal shell model), and on the right, we show the active shells at 𝑁=20and 𝑁=28inside the coloured blocks. A change in the order of the shells leads to shell inversion. the 𝑁=28shell closure for potential two-neutron Borromean halos in the Na and Mg isotopes (as shown in the blue circles in the upper panel of Fig. 1). The magnesium isotopic chain also provides interesting candidates to explore the transition from strongly-bound to weakly-bound nuclei and the form of 𝑛-𝑛correlations at the limits of stability. Deformation will also play a role: e.g., the nucleus 32Mg, with 𝑁=20, can be described as a prolate-deformed rotor [8,33]. There is only limited knowledge of the properties of heavier neutron-rich Mg isotopes with 𝑁≥25. The weakly-bound nucleus 37Mg (𝑁=25) has been observed, and its ground state can be interpreted as a one-neutron halo with the valence neutron occupying 2𝑝3∕2 orbit [34–38]. The neighbouring odd- A nucleus 39Mg was confirmed as unbound, indicating the important role of pair correlations. For the present study, we start from the medium-mass open shell nucleus 40Mg, which is the next two-neutron halo candidate after 29F according to various theoretical models [16,39–41]. The evaluated (i.e., not measured but estimated based on systematics) two-neutron separation energy of 40Mg is 𝑠2𝑛=0.670 ±0.710 MeV [42]. This nucleus was first observed experimentally in 2007 [43]and more recently low-lying excited states of 40Mg have been studied using a one-proton removal reaction from 41Al [12]. Recent ab initio calculations suggest large deformation in 40Mg [44], though no connection to halo physics is made. Because 39Mg is neutron unbound, it is reasonable to assume that the correlation between the two valence neutrons in 40Mg plays an important role in binding this system. Thus, the 40Mg nucleus, when interpreted as a 38Mg core plus two valence neutrons, provides an example of a Borromean system, similar to the well-studied two-neutron halo nuclei 6He [45,46]or 11Li [47]. Because three-body models give a good description of the structural properties of many Borromean nuclei, we apply the same technique, using effective three-body calculations to study the configuration mixing in and matter radius of the ground state of 40Mg. Given the uncertainties in the two-neutron separation energy, we explore how the allowed values influence the configuration mixing and matter radius, by analysing different scenarios. This allows us to explore the possibility of halo formation in this region of the nuclear chart, well beyond the heaviest observed two-neutron halo in 29F[13,30–32]. The most recent atomic mass evaluation [42] indicates that 39Na is unbound, contradicting Ahn et al. [26] recent experimental discovery; our calculations within same three-body framework aim to predict the two-neutron separation energy (𝑠2𝑛) and matter radii for 39Na systematically. Finally, we also provide Glauber-model predictions for the total reaction cross sections of 40Mg and 39Na on a carbon target, which will further support the finding of halo formation in these systems. 2. Model formulation To focus on the effective core +𝑛 +𝑛system, we employ the hyperspherical formalism [48,49]. The eigenstates of the three-body Hamiltonian can be written as Ψ(𝜌,Ω) = 𝜌−5∕2 ∑ 𝛽 𝑈𝛽(𝜌)𝛽(Ω),(1) where 𝜌2=𝑥2+𝑦2is the hyperradius defined from the usual massscaled Jacobi coordinates {𝒙,𝒚}, and the hyperangle 𝛼=arctan(𝑥∕𝑦) is contained in the angular variables Ω ={𝛼,𝑥, 𝑦}. For simplicity, we choose 𝒙as the coordinate describing the neutron-neutron relative motion, and 𝒚as the coordinate between the core and the two neutrons. This is typically referred to as the Jacobi-𝑇representation. In Eq. (1), the index 𝛽≡{𝐾, 𝑙𝑥, 𝑙𝑦, 𝑙, 𝑆𝑥}𝑗labels the different components of the wave function, with 𝛽(Ω) representing states of total angular momentum 𝑗which follow the coupling order 𝛽(Ω) = [Υ𝑙𝑥𝑙𝑦 𝐾𝑙 (Ω) ⊗𝜅 𝑆𝑥]𝑗𝜇 .(2) Here, Υ𝑙𝑥𝑙𝑦 𝐾𝑙𝑚𝑙(Ω) are the hyperspherical harmonics [48], eigenstates of the hypermomentum operator  𝐾, and 𝜅𝑆𝑥is a spin function. This equation implies that 𝒍=𝒍𝑥+𝒍𝑦is the total orbital angular momentum, 𝑆𝑥 is the coupled spin of the two neutrons, and the total 𝑗results from 𝒋=𝒍+𝑺𝑥. Note that we assume the core to be inert and its spin to be zero. More details can be found, for instance, in Ref. [50]. To determine the hyperradial functions 𝑈𝛽(𝜌)for bound states, we diagonalize the Hamiltonian in a suitable basis and focus on the negative-energy solutions. Thus, we expand 𝑅𝛽(𝜌)= 𝑛 ∑ 𝑖=0 𝐶𝑖𝛽 𝑈𝑖𝛽 (𝜌),(3) where 𝐶𝑖𝛽 are the diagonalization coefficients to be determined numerically. For this purpose, different bases have been used in various works [51–53], and in the present work, we employ the analytical transformed harmonic oscillator (THO) basis from Ref. [54]. The THO functions are built by performing a local scale transformation to the harmonic oscillator functions, changing their Gaussian asymptotic behaviour into an exponential decay, which is better suited to accurately describe bound nuclear states with a relatively small set of basis functions. This procedure was also adopted, for instance, in Refs. [30,32]. The diagonalization of the three-body Hamiltonian requires the computation of the corresponding kinetic energy and potential matrix elements. With the above definition in hyperspherical coordinates, we can write [55,56] 𝑇𝛽(𝜌)=−ℏ2 2𝑚(𝑑2 𝑑𝜌2−15∕4 + 𝐾(𝐾+4) 𝜌2),(4) for the kinetic energy operator, where 𝑚is the mass of the nucleon. The coupling potentials are taken to be of the form 𝑉𝛽𝛽′(𝜌)=⟨𝛽(Ω)|||𝑉12 +𝑉13 +𝑉23|||𝛽′(Ω)⟩+𝛿𝛽𝛽′𝑉3b(𝜌).(5) Physics Letters B 853 (2024) 138694 3 J. Singh, J. Casal, W. Horiuchi et al. In this expression, 𝑉𝑖𝑗 are the two-body interactions within the threebody composite system, to be fixed from the known information on the binary subsystems, and 𝑉3b(𝜌)is a phenomenological three-body force, to account for effects not included in the two-body interactions alone [53,54,56,57]. This term contains the only free parameter in this model and is used to adjust the energy of the three-body states to the experimental values if known. 3. Two-body (core +𝒏) subsystems The spectral properties of the core +𝑛subsystems play a key role in the study of the structure of Borromean three-body nuclei, as the two-body potentials are contained in the three-body Hamiltonian. In the present case, this amounts to fixing a phenomenological 38Mg + 𝑛 and 37Na + 𝑛potential to describe the low-lying continuum spectrum of 39Mg and 38Na, respectively. We will start with 38Mg + 𝑛system. Though 𝑁=26has a partially filled 1𝑓7∕2 subshell, we model 39Mg as a 38Mg core surrounded by an unbound neutron moving in the 𝑓7∕2, 𝑝3∕2, 𝑓5∕2, and 𝑝1∕2 orbitals in a simple independent-particle shell-model picture. Even though the distinction between the core and valence neutrons is not completely obvious, for simplicity, we nevertheless make an inert-core approximation. This means that the effects from internal rearrangement or core-valence exchange must be contained in effective potential parameters. Note that a similar approach has been successfully followed in other three-body calculations, such as the 27F+𝑛and the 29F+𝑛subsystems for the description of 29F[30–32]and 31F[58]. Core excitations could be incorporated by using, for instance, a particle-rotor model [41] with an effective core deformation parameter. The only theoretical study on the spectrum of the neutron-unbound nucleus 39Mg reports an indication of a low-lying resonance at 0.129 MeV with 𝐽𝜋either 7∕2−or 3∕2−[59]. However, due to the unavailability of experimental data, this does not rule out the existence of other possibilities such as a single resonant structure or more than two states, and in any case, the spin-parity assignment is not unambiguous. Considering the limited information available, we model the 38Mg + 𝑛interaction as a Woods-Saxon potential including only central and spin-orbit terms, 𝑉core+𝑛(𝑟)=(−𝑉(𝑙) 0+𝑉𝑙𝑠𝜆2 𝜋𝒍⋅𝒔1 𝑟 𝑑 𝑑𝑟)1 1+exp[(𝑟−𝑅𝑐)∕𝑎],(6) where 𝑉𝑙 0is in general 𝑙-dependent, and 𝑅𝑐=𝑟0𝐴1∕3 𝑐with 𝐴𝑐the mass number of the core. The spin-orbit interaction is written in terms of the Compton wavelength 𝜆𝜋=1.414 fm. Following Ref. [60], the spinorbit strength is taken to follow the systematic trend [61]and has the value 𝑉𝑙𝑠 = 16.842 MeV for 38Mg + 𝑛and 16.324 MeV for 37Na + 𝑛. The value 𝑟0=1.25 fm is as originally suggested for 31Ne [30,60], and we examine three scenarios, see Table 1. In Set 1and 2, 𝑉0is chosen to be 𝑙independent. In set 1it is adjusted to fix the 𝑓7∕2 ground-state resonance of 39Mg at 0.129 MeV, corresponding to the prediction of Ref. [59]. This set gives a “normal” shell-model scenario, with an additional 𝑝3∕2 resonance appearing at higher energy, 0.349 MeV. To explore the competition of the 𝑝-wave with the 𝑓-wave ground state, we consider two additional scenarios defined by Sets 2and 3, respectively. In the “degenerate” case, we tune the 𝑓-wave strength and diffuseness parameter 𝑎so that the 𝑓7∕2 and 𝑝3∕2 resonances become nearly degenerate. The “inverted” scenario assumes the ground state of 39Mg to be the 𝑝-wave resonance. Here we need to allow orbital dependence, and we have different potential depths for 𝑙=1and 𝑙=3. These potentials generate Pauli forbidden single-particle states, which would result in unphysical eigenstates of the three-body Hamiltonian, and need to be removed. In this work, this is achieved through a supersymmetric transformation [62,63]. As we do not have either theoretical or experimental predictions and data for 38Na, we use the same potential parameters as for 39Mg. The only changes are 𝑅𝑐and the spin-orbit strength, as explained above. Table 1 Parameter sets for the 38Mg + 𝑛(upper-panel) and 37Na + 𝑛(lowerpanel) Woods-Saxon interactions, Eq. (6). Here 𝑎is diffuseness, 𝑉(𝑙) 0is the potential depth and 𝐸𝑅is the position of the resonances. Note that 𝑟0=1.25 fm, and 𝑉𝑙𝑠 =16.842 MeV (for 38Mg + 𝑛) and 16.324 MeV (for 37Na + 𝑛) are fixed. 38Mg + 𝑛 Set Scenario 𝑎(fm) 𝑙𝑗 𝑉 (𝑙) 0(MeV) 𝐸𝑅(MeV) 1Normal 0.70 𝑓7∕2 38.225 0.129 𝑝3∕2 38.225 0.349 2Degenerate 0.75 𝑓7∕2 38.400 0.129 𝑝3∕2 38.400 0.135 3Inverted 0.75 𝑓7∕2 37.880 0.349 𝑝3∕2 38.425 0.130 37Na + 𝑛 1Degenerate 0.70 𝑓7∕2 38.225 0.539 𝑝3∕2 38.225 0.599 2Inverted 0.75 𝑓7∕2 38.400 0.522 𝑝3∕2 38.400 0.271 3Inverted 0.75 𝑓7∕2 37.880 0.734 𝑝3∕2 38.425 0.265 The latter takes into account effectively the tensor force between the unpaired proton and valence neutron [64,65]. Indeed, since 39Na has an odd proton number, its ground state angular momentum is not zero. However, we have found before [30,58,66]that approximating the ground state of these light systems as a 0+state gives reasonable results. The lower half of Table 1shows the parameter sets for the 37Na+𝑛 potential. Our predictions for the low-lying spectrum of 38Na show no “normal” ordering of the single-particle states; most scenarios present an inversion, indicating a shell-evolution effect, that we describe effectively by just changing the spin-orbit strength. 4. Three-body (core +𝒏 +𝒏) calculations Using the core+𝑛potentials outlined in the preceding section, we compute the three-body ground-state of the 40Mg and 39Na nuclei assuming a simple description in which two valence neutrons couple to 𝐽𝜋=0 +. To provide a more comprehensive description, it is essential to have precise experimental data on the core+𝑛spectrum and a thorough understanding of the intricate core+𝑛interactions that can result in the splitting of single-particle energy levels. In the present calculation, we also need a prescription for the 𝑛𝑛 interaction. We use the Gogny- Pires-Tourreil (GPT) potential [67]as in previous works [54,56,68]. It is important to emphasize that when it comes to the ground-state properties of core +𝑛 +𝑛systems, the specific selection of the 𝑛𝑛 interaction is not particularly critical, as long as the chosen interaction adequately represents 𝑛𝑛 scattering data, as discussed in Ref. [48]. As previously introduced in Eq. (5), we incorporate a phenomenological three-body force to obtain the energy of the three-body ground state, following a methodology similar to that employed in our earlier studies [30,54,58]. The three-body problem is solved by diagonalizing the Hamiltonian in a THO basis including hyperspherical harmonics up to a maximum hypermomentum 𝐾𝑚𝑎𝑥 in the wave-function expansion (Eq. (1)), and 𝑖 =0, … , 𝑁excitations for the hyperradial functions (Eq. (3)). Note that fixing 𝐾𝑚𝑎𝑥 implies that the orbital angular momenta associated with each Jacobi coordinate are restricted to 𝑙𝑥+𝑙𝑦≤𝐾[48]. In this work, 𝐾𝑚𝑎𝑥 =46and 𝑁=30are found to provide converged results. In order to compute the 40Mg and 39Na matter radius within the present three-body model [55]using 𝑅𝑚=√1 𝐴(𝐴𝑐𝑅2 𝑐+⟨𝑟2 𝑛𝑛⟩), the size of the 38Mg and 37Na cores is required as input. We have used 𝑅𝑐= 𝑅𝑚(38Mg,37 Na) =3.600, 3.637 fm, respectively, adopted from Ref. [69, 70]. These are somewhat uncertain, so it is more precise to look at a Physics Letters B 853 (2024) 138694 4 J. Singh, J. Casal, W. Horiuchi et al. Fig. 2. (upper panel) The occupancy of the (𝑓7∕2)2(solid lines) and (𝑝3∕2)2or- bits (dashed lines) for different parameter sets in 40Mg (left) and 39Na (right), respectively, as a function of 𝑠2𝑛. (lower panel) Variation of matter radii (R𝑚) for 40Mg and 39Na with 𝑠2𝑛. The black dotted line shows the core radius, 38Mg and 37Na, respectively. It should be noted that the labels Inverted, Degenerate and Normal pictures here refer to the outcome of full three-body calculations and should not be confused with the single-particle scenarios of Table 1. comparison of the differences across various potential models as well as the relative increase between core and three-body systems. With all these ingredients, our three-body results for the ground state of 40Mg and 39Na using the scenarios for the core+𝑛potentials are shown in Fig. 2. As already discussed, there are large uncertainties in the evaluated 𝑠2𝑛value [42]of 40Mg and no information is available for the 𝑠2𝑛value of 39Na. Given these uncertainties, we explore the sensitivity of the configuration mixing and matter radius of the ground state with 𝑠2𝑛, using different values of the three-body potential strength. For 40Mg, 𝑉3𝑏=0leads to an overbinding of the system, so we need to choose 𝑉3𝑏>0to get 𝑠2𝑛corresponding to the lower, central and upper limits of the evaluation (0.010, 0.670 and 1.380 MeV). Note that the lower limit would result in an unbound state. However, the experimental findings support a bound ground state, so we have considered a barely bound case as well. For 39Na, due to the lack of experimental information and systematic information on 𝑠2𝑛, it seems reasonable to keep the range of 𝑉3𝑏values determined for 40Mg and make predictions. With this prescription, the shallowest case becomes unbound for all sets. To restrict the situation to bound states, we modify 𝑉3𝑏in that case so that 𝑠2𝑛is 0.010 MeV, since we are interested in giving predictions for a bound halo nucleus, which could be tested experimentally. We predict 𝑠2𝑛for 39Na between 0.010-0.824 (1.828)MeV without (with) 𝑉3𝑏=0cases. The computed partial wave content and radii for each 𝑠2𝑛value are shown in the upper panel and lower panel, respectively, of Fig. 2. As expected, shallower (bound) ground states yield larger 𝑝-wave occupancy which in turn leads to an “inverted” picture, in contrast to the deeply-bound ground states which lead to the “degenerate” and “normal” pictures for all sets. As can be seen from the upper panel of Fig. 2, when moving from Set 1 to Set 3 the 𝑝-wave content in the ground state increases, whereas the 𝑓-wave content decreases for each choice of 𝑠2𝑛 value. We remark that the similar behaviour was found in Ref. [71]. Similarly, we see in the lower panel of Fig. 2that, as expected, the shallowest cases give larger matter radii in contrast with more deeply bound ground states, which result in smaller matter radii for all sets in both systems considered. More interestingly, we note that the effect of the value of 𝑠2𝑛on the change in radius is most substantial when we have a shell inversion. However, due to the rather heavy cores, the calculated total radii are similar for the scenarios considered. Even with a substantial neutron halo, for a fixed 𝑠2𝑛the variations amount up to a Fig. 3. The Variation of matter radii (𝑅𝑚) for Mg (upper panel) and Na (lower panel) isotopes with mass number 𝐴. Experimental values, shown in black, blue, and red circles, were taken from Refs. [72], [73], and [34], respectively. The dotted black line corresponds to the 𝑅0𝐴1∕3 fit for 𝐴 =24-37 for Mg-isotopes and 𝐴 =23-32 for Na-isotopes. It should be noted that each colour band associated with a specific 𝑠2𝑛corresponds to the range of values for Sets 1, 2, and 3, and the same applies to Fig. 5. maximum of 5% difference for both systems. Differentiating scenarios solely by radius could pose experimental difficulties. Additionally, data on knockout or transfer reactions that can reveal the partial wave content in the ground state of these nuclei would be very useful to tighten the constraints on theoretical models and enable us to distinguish between various wave functions that have been presented in this text. In the lower panel of Fig. 2, we show the radius of the 38Mg and 37Na cores as a horizontal dotted black line which makes it easy to extract the relative change in matter radius of 40Mg and 39Na relative to the radius of the appropriate core, Δ𝑅 =𝑅𝑚−𝑅𝑐𝑜𝑟𝑒. With the current uncertainties in 𝑠2𝑛and the low-lying spectrum of subsystems, the difference of the matter radius of these nuclei and their cores ranges between 0.1-0.5fm for different choices of potential sets and 𝑠2𝑛values. This number is notably smaller than those for well-established halo nuclei such as 6He or 11Li, but is close to the heaviest known two-neutron halo, 29F, which is only 0.35 fm larger than the 27Fcore [13,30,32]. It is worth mentioning that, recently, we have shown that the halo formation in 29Fis connected to the weakening of the 𝑁=20shell gap, which leads to the intrusion of the 𝑝3∕2 orbital and reduced binding [30–32], which leads to halo formation. The same situation is observed when 𝑁=28 melts, with the difference that the 𝑝3∕2 orbital is within the same major shell. As depicted in Fig. 2, our three-body findings reveal a substantial increase in Δ𝑅for the least deeply bound state, with a substantial 𝑝3∕2 component. This suggests that our study provides evidence for a halo structure in the ground states of both 40Mg and 39Na. However, experimental cross section measurements are necessary to confirm this assertion. In Fig. 3we show the variation of matter radii for the Mg and Na isotopes with increasing mass numbers. The experimental derived values for the radii for 20−32Mg and 20−32Na (shown in black circles) are adopted from Ref. [72], while those for 32−25Mg are taken from Ref. [73](shown in blue circles). Data for the one-neutron halo 37Mg (shown as a red circle) is taken from Ref. [34]. The dotted black lines in the figure correspond to a weighted fit of the experimental data points of the form 𝑅0𝐴1∕3 for 𝐴 =24–37 for Mg isotopes and 𝐴 =23–32 for Na isotopes, respectively. It must be noted that the sharp rise of radius in 37Mg compared to the nearest lighter isotope can be attributed to the fact that 37Mg is itself a one-neutron halo nucleus [34]. It can be easily seen from Fig. 3that the radii of 40Mg and 39Na are larger than the standard 𝑅0𝐴1∕3 fitted value for all choices of 𝑠2𝑛. The present calculations provide evidence of a modest halo in all instances, with a more pronounced effect observed as the occupation of the 𝑝3∕2 intruder or- Physics Letters B 853 (2024) 138694 5 J. Singh, J. Casal, W. Horiuchi et al. Fig. 4. The ground state probability density distribution (in units of fm−2) for 40Mg for Set-3 (a) with 𝑠2𝑛=0.010 MeV and (b) with 𝑠2𝑛=1.380 MeV. The contour lines in (b) show the shape of the peak above a probability of 0.04, which is twice as high as that in (a). bital increases, akin to the case of 29F. Thus, this observation implies a likely two-neutron halo structure in the ground state of 40Mg and 39Na, and the corresponding melting of the traditional 𝑁=28shell gap is due to the intrusion of the 𝑝3∕2 orbital. We can gain a more in-depth understanding of the wave function by examining the probability densities in the Jacobi-T system, scaling 𝑥and 𝑦to the distance between the valence neutrons (𝑟𝑛𝑛) and that between the centre of mass of the neutrons and the core (𝑟𝑐−𝑛𝑛). Fig. 4illustrates density distributions for 40Mg in the inverted scenario, considering both lower and upper limits of 𝑠2𝑛, which encourages halo formation in its ground state. The density distributions for 39Na closely resemble those of 40Mg and are consequently not shown. Both selections of the 𝑠2𝑛result in a peak where two neutrons are situated near each other at a distance from the core, the di-neutron peak. However, the latter choice leads to a more concentrated wave function, which is a consequence of the larger 𝑓-wave component (evidenced by the presence of four peaks in Fig. 4b). It is also evident that the valence neutrons explore shorter relative distances. These characteristics are reflected in the root-mean-square (rms) values of the coordinates, with values of 11.35 and 8.06 fm for 𝑟𝑛𝑛 and 𝑟𝑐−𝑛𝑛, respectively, in the shallow case, and 6.43 and 4.55 fm in the deeper case. In summary, the results presented in this section provide evidence for the formation of halos in the ground state of both 40Mg and 39Na systems. The formation of these halos is attributed to the dissolution of the 𝑁=28shell gap, leading to the intrusion of the 𝑝3∕2 orbital and subsequent weak binding. 5. Total reaction cross section Experimentally, a very obvious way to determine whether a nucleus is a halo nucleus, is to look for an enhanced reaction cross section. Thus we examine the total reaction cross section (𝜎𝑅) by employing the conventional Glauber theory [74]. We utilize the nucleontarget formalism [75]and apply the nucleon-nucleon profile function from Refs. [76–78]. This approach has proven effective in various high-energy nucleus-nucleus collision reactions, particularly those with unstable nuclei, and it successfully replicated isotope dependence of the total reaction cross sections with appropriate density distributions [60,79–84]. The additional theoretical inputs required for this reaction model include the density distributions of both the projectile and target nuclei. In the case of 40Mg, we first generate a harmonic-oscillator (HO) type density distribution for 38Mg that reproduces the measured total reaction cross section of 38Mg+12Cby Takechi et al., [34]at 240 MeV/nucleon (1535 ±21 mb) by assuming a simple shell model state (1𝑓7∕2)8 𝜈⊗(1𝑑5∕2)4 𝜋. The density for 40Mg was constructed by simply adding the 2𝑛densities calculated in the previous section to the 38Mg density. No centre-of-mass correction was applied in these calculations, but we believe this is a safe strategy for a system with such a heavy Fig. 5. Measured and calculated total reaction cross section for different isotopes of Mg and Na at different energies. Theoretical results (shown with green, magenta, and indigo bars) for 40Mg+12Care shown at (a) 240 MeV/A and (b) 1000 MeV/A. Experimental data shown with red, blue, and black dots are taken from [34](at 240 MeV/A), [73](at 900 MeV/A), and [72](at 950 MeV/A), respectively. (c) Shows the results for 39Na+12Cat 1000 MeV/A and experimental data shown with black dots is taken from [72](at 950 MeV/A). The dotted black line corresponds to a 𝜎0𝐴2∕3 fit to guide the eye of the reader. core. Using this prescription, we predict the 𝜎𝑅for 40Mg at different incident energies, 240 MeV/nucleon (lies between 1601-1807 mb, shown in Fig. 5(a)) and 1000 MeV/A (lies between 1695-1944 mb, shown in the Fig. 5(b)). As can be seen from Fig. 5(a) and (b), the predicted values of 𝜎𝑅for 40Mg show significant enhancement with respect to the observed 𝜎𝑅in the lower-𝐴isotopes for both choices of energy. This shows that we can use these reaction cross sections as an indication of the melting of the 𝑁=28shell closure. We use the same HO parameterization established for 38Mg for 37Na. Hence, we forecast the 𝜎𝑅for 39Na at an energy of 1000 MeV/nucleon, as depicted in Fig. 5(c) (lies between 1710-1894 mb). Our computed estimates of 𝜎𝑅for 39Na reveal a noteworthy increase compared to the observed 𝜎𝑅in its lower-mass isotopes. This reinforces our earlier finding in the 40Mg, which points to the formation of a two-neutron halo in the ground state of 39Na, leading to the disruption of the neutron shell at 𝑁=28. 6. Conclusions We have analysed the configuration mixing and matter radius of the ground states of the nuclei 40Mg and 39Na at the potential 𝑁=28shell closure. Since we are using an effective three-body model, we can study two-neutron halo formation. Unfortunately, there is very little information on the properties of 39Mg, which we would need to constrain our models. We have instead analysed three scenarios that span the range of allowed parameters. We have then used the same parameters for both 40Mg and 39Na. We have thus mapped out the whole range of the expected 𝑠2𝑛values in 40Mg and 39Na. Our results indicate a picture with mild to strong mixing of the intruder 𝑝3∕2 orbit with the normal 𝑓7∕2 orbit due to the magnitude of the core +𝑛potentials in our three scenarios. This seems difficult to avoid in 39Na, where most scenarios lead to a shell inversion. It seems only plausible for 40Mg, where a normal ordering is also possible. The mixing enhances the di-neutron configuration in the ground-state density. Due to the heavy masses of these nuclei, the computed matter radii vary only up to 5% for a fixed 𝑠2𝑛and considering the different choices of the core+𝑛interaction. This makes it challenging to verify Physics Letters B 853 (2024) 138694 6 J. Singh, J. Casal, W. Horiuchi et al. this experimentally, since one must make highly accurate mass measurements. On the theory side, we need a thorough understanding of the low-energy continuum spectrum of core+𝑛subsystems. Additionally, the inclusion of transfer or knockout data, capable of probing the partial-wave characteristics, would be highly valuable in distinguishing between the scenarios used in this study. The matter radii of 40Mg and 39Na show an increase of 0.1to 0.5fm compared to their respective cores 38Mg and 37Na, lending support to the idea of a potential halo structure in their ground state. This assertion gains further backing from the elevated predicted reaction cross sections for these nuclei in comparison to the lower mass isotopes. Nevertheless, it is imperative to verify this conclusion through experimental measurements of interaction cross sections. Future perspectives include incorporating core deformation within the effective three-body description explicitly, to assess its influence on the halo structure in this region of the nuclear chart. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Data availability Data will be made available on request. Acknowledgements We thank L. Fortunato for careful reading of the manuscript. This work was supported by the UK Science and Technology Funding Council [grant number ST/V001116/1] (JS and NRW), by the Polish National Science Centre (NCN) [under Contract No. 2018/31/B/ST2/02220] (JS and WS), by the Spanish MCIN/AEI/10.13039/501100011033 [under I+D+i project No. PID2020-114687GB-I00] (JC), and by the JSPS KAKENHI Grants Nos. 18K03635 and 22H01214 (WH). References [1] E.K. Warburton, J.A. Becker, B.A. Brown, Mass systematics for a=29–44 nuclei: the deformed a∼32 region, Phys. Rev. C 41 (1990) 1147–1166, https://doi .org /10 . 1103 /PhysRevC .41 .1147, https://link .aps .org /doi /10 .1103 /PhysRevC .41 .1147. [2] J. Dobaczewski, I. Hamamoto, W. Nazarewicz, J.A. Sheikh, Nuclear shell structure at particle drip lines, Phys. Rev. Lett. 72 (1994) 981–984, https://doi .org /10 .1103 / PhysRevLett .72 .981, https://link .aps .org /doi /10 .1103 /PhysRevLett .72 .981. [3] O. Sorlin, M.-G. Porquet, Nuclear magic numbers: new features far from stability, Prog. Part. Nucl. Phys. 61 (2) (2008) 602–673, https://doi .org / 10 .1016 /j .ppnp .2008 .05 .001, https://www .sciencedirect .com /science /article /pii / S0146641008000380. [4] K. Heyde, J.L. Wood, Shape coexistence in atomic nuclei, Rev. Mod. Phys. 83 (2011) 1467–1521, https://doi .org /10 .1103 /RevModPhys .83 .1467, https://link . aps .org /doi /10 .1103 /RevModPhys .83 .1467. [5] T. Otsuka, Y. Tsunoda, The role of shell evolution in shape coexistence, J. Phys. G, Nucl. Part. Phys. 43 (2) (2016) 024009, https://doi .org /10 .1088 /0954 -3899 /43 /2 / 024009. [6] H. Nishibata, S. Kanaya, T. Shimoda, A. Odahara, S. Morimoto, A. Yagi, H. Kanaoka, M.R. Pearson, C.D.P. Levy, M. Kimura, N. Tsunoda, T. Otsuka, Structure of 31Mg: shape coexistence revealed by 𝛽−𝑔𝑎𝑚𝑚𝑎 spectroscopy with spin-polarized 31Na, Phys. Rev. C 99 (2019) 024322, https://doi .org /10 .1103 /PhysRevC .99 .024322, https://link .aps .org /doi /10 .1103 /PhysRevC .99 .024322. [7] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi, N. Takahashi, Measurements of interaction cross sections and nuclear radii in the light 𝑝-shell region, Phys. Rev. Lett. 55 (1985) 2676–2679, https://doi .org /10 .1103 /PhysRevLett .55 .2676, https://link . aps .org /doi /10 .1103 /PhysRevLett .55 .2676. [8] T. Motobayashi, Y. Ikeda, K. Ieki, M. Inoue, N. Iwasa, T. Kikuchi, M. Kurokawa, S. Moriya, S. Ogawa, H. Murakami, S. Shimoura, Y. Yanagisawa, T. Nakamura, Y. Watanabe, M. Ishihara, T. Teranishi, H. Okuno, R. Casten, Large deformation of the very neutron-rich nucleus 32mg from intermediate-energy Coulomb excitation, Phys. Lett. B 346 (1) (1995) 9–14, https://doi .org /10 .1016 /0370 -2693(95 )00012 - A, https://www .sciencedirect .com /science /article /pii /037026939500012A. [9] T. Nakamura, N. Kobayashi, Y. Kondo, Y. Satou, N. Aoi, H. Baba, S. Deguchi, N. Fukuda, J. Gibelin, N. Inabe, M. Ishihara, D. Kameda, Y. Kawada, T. Kubo, K. Kusaka, A. Mengoni, T. Motobayashi, T. Ohnishi, M. Ohtake, N.A. Orr, H. Otsu, T. Otsuka, A. Saito, H. Sakurai, S. Shimoura, T. Sumikama, H. Takeda, E. Takeshita, M. Takechi, S. Takeuchi, K. Tanaka, K.N. Tanaka, N. Tanaka, Y. Togano, Y. Utsuno, K. Yoneda, A. Yoshida, K. Yoshida, Halo structure of the island of inversion nucleus 31Ne, Phys. Rev. Lett. 103 (2009) 262501, https://doi .org /10 .1103 /PhysRevLett . 103 .262501, https://link .aps .org /doi /10 .1103 /PhysRevLett .103 .262501. [10] M. Takechi, T. Ohtsubo, M. Fukuda, D. Nishimura, T. Kuboki, T. Suzuki, T. Yamaguchi, A. Ozawa, T. Moriguchi, H. Ooishi, D. Nagae, H. Suzuki, S. Suzuki, T. Izumikawa, T. Sumikama, M. Ishihara, H. Geissel, N. Aoi, R.-J. Chen, D.-Q. Fang, N. Fukuda, I. Hachiuma, N. Inabe, Y. Ishibashi, Y. Ito, D. Kameda, T. Kubo, K. Kusaka, M. Lantz, Y.-G. Ma, K. Matsuta, M. Mihara, Y. Miyashita, S. Momota, K. Namihira, M. Nagashima, Y. Ohkuma, T. Ohnishi, M. Ohtake, K. Ogawa, H. Sakurai, Y. Shimbara, T. Suda, H. Takeda, S. Takeuchi, K. Tanaka, R. Watanabe, M. Winkler, Y. Yanagisawa, Y. Yasuda, K. Yoshinaga, A. Yoshida, K. Yoshida, Interaction cross sections for ne isotopes towards the island of inversion and halo structures of 29ne and 31ne, Phys. Lett. B 707 (3) (2012) 357–361, https://doi .org /10 .1016 /j .physletb .2011 .12 . 028, https://www .sciencedirect .com /science /article /pii /S0370269311014857. [11] I. Hamamoto, Neutron shell structure and deformation in neutron-drip-line nuclei, Phys. Rev. C 85 (2012) 064329, https://doi .org /10 .1103 /PhysRevC .85 .064329, https://link .aps .org /doi /10 .1103 /PhysRevC .85 .064329. [12] H.L. Crawford, P. Fallon, A.O. Macchiavelli, P. Doornenbal, N. Aoi, F. Browne, C.M. Campbell, S. Chen, R.M. Clark, M.L. Cortés, M. Cromaz, E. Ideguchi, M.D. Jones, R. Kanungo, M. MacCormick, S. Momiyama, I. Murray, M. Niikura, S. Paschalis, M. Petri, H. Sakurai, M. Salathe, P. Schrock, D. Steppenbeck, S. Takeuchi, Y.K. Tanaka, R. Taniuchi, H. Wang, K. Wimmer, First spectroscopy of the near drip-line nucleus 40Mg, Phys. Rev. Lett. 122 (2019) 052501, https://doi .org /10 .1103 /PhysRevLett . 122 .052501, https://link .aps .org /doi /10 .1103 /PhysRevLett .122 .052501. [13] S. Bagchi, R. Kanungo, Y.K. Tanaka, H. Geissel, P. Doornenbal, W. Horiuchi, G. Hagen, T. Suzuki, N. Tsunoda, D.S. Ahn, H. Baba, K. Behr, F. Browne, S. Chen, M.L. Cortés, A. Estradé, N. Fukuda, M. Holl, K. Itahashi, N. Iwasa, G.R. Jansen, W.G. Jiang, S. Kaur, A.O. Macchiavelli, S.Y. Matsumoto, S. Momiyama, I. Murray, T. Nakamura, S.J. Novario, H.J. Ong, T. Otsuka, T. Papenbrock, S. Paschalis, A. Prochazka, C. Scheidenberger, P. Schrock, Y. Shimizu, D. Steppenbeck, H. Sakurai, D. Suzuki, H. Suzuki, M. Takechi, H. Takeda, S. Takeuchi, R. Taniuchi, K. Wimmer, K. Yoshida, Two-neutron halo is unveiled in 29F, Phys. Rev. Lett. 124 (2020) 222504, https://doi .org /10 .1103 /PhysRevLett .124 .222504, https://link .aps .org /doi /10 .1103 /PhysRevLett .124 .222504. [14] T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, Y. Utsuno, Evolution of shell structure in exotic nuclei, Rev. Mod. Phys. 92 (2020) 015002, https://doi .org /10 . 1103 /RevModPhys .92 .015002, https://link .aps .org /doi /10 .1103 /RevModPhys .92 . 015002. [15] W. Horiuchi, T. Inakura, S. Michimasa, Large enhancement of total reaction cross sections at the edge of the island of inversion in ti, cr, and fe isotopes, Phys. Rev. C 105 (2022) 014316, https://doi .org /10 .1103 /PhysRevC .105 .014316, https://link . aps .org /doi /10 .1103 /PhysRevC .105 .014316. [16] E. Caurier, F. Nowacki, A. Poves, Merging of the islands of inversion at 𝑛 =20 and 𝑛 =28, Phys. Rev. C 90 (2014) 014302, https://doi .org /10 .1103 /PhysRevC . 90 .014302, https://link .aps .org /doi /10 .1103 /PhysRevC .90 .014302. [17] P. Doornenbal, H. Scheit, S. Takeuchi, N. Aoi, K. Li, M. Matsushita, D. Steppenbeck, H. Wang, H. Baba, H. Crawford, C.R. Hoffman, R. Hughes, E. Ideguchi, N. Kobayashi, Y. Kondo, J. Lee, S. Michimasa, T. Motobayashi, H. Sakurai, M. Takechi, Y. Togano, R. Winkler, K. Yoneda, In-beam 𝛾-ray spectroscopy of 34,36,38Mg: merging the 𝑛=20 and 𝑛=28 shell quenching, Phys. Rev. Lett. 111 (2013) 212502, https://doi .org / 10 .1103 /PhysRevLett .111 .212502, https://link .aps .org /doi /10 .1103 /PhysRevLett . 111 .212502. [18] H. Scheit, T. Glasmacher, B.A. Brown, J.A. Brown, P.D. Cottle, P.G. Hansen, R. Harkewicz, M. Hellström, R.W. Ibbotson, J.K. Jewell, K.W. Kemper, D.J. Morrissey, M. Steiner, P. Thirolf, M. Thoennessen, New region of deformation: the neutron-rich sulfur isotopes, Phys. Rev. Lett. 77 (1996) 3967–3970, https://doi .org /10 .1103 / PhysRevLett .77 .3967, https://link .aps .org /doi /10 .1103 /PhysRevLett .77 .3967. [19] T. Glasmacher, B. Brown, M. Chromik, P. Cottle, M. Fauerbach, R. Ibbotson, K. Kemper, D. Morrissey, H. Scheit, D. Sklenicka, M. Steiner, Collectivity in 44s, Phys. Lett. B 395 (3) (1997) 163–168, https://doi .org /10 .1016 /S0370 -2693(97 )00077 -4, https://www .sciencedirect .com /science /article /pii /S0370269397000774. [20] S. Takeuchi, M. Matsushita, N. Aoi, P. Doornenbal, K. Li, T. Motobayashi, H. Scheit, D. Steppenbeck, H. Wang, H. Baba, D. Bazin, L. Càceres, H. Crawford, P. Fallon, R. Gernhäuser, J. Gibelin, S. Go, S. Grévy, C. Hinke, C.R. Hoffman, R. Hughes, E. Ideguchi, D. Jenkins, N. Kobayashi, Y. Kondo, R. Krücken, T. Le Bleis, J. Lee, G. Lee, A. Matta, S. Michimasa, T. Nakamura, S. Ota, M. Petri, T. Sako, H. Sakurai, S. Shimoura, K. Steiger, K. Takahashi, M. Takechi, Y. Togano, R. Winkler, K. Yoneda, Well developed deformation in 42Si, Phys. Rev. Lett. 109 (2012) 182501, https://doi .org / 10 .1103 /PhysRevLett .109 .182501, https://link .aps .org /doi /10 .1103 /PhysRevLett . 109 .182501. [21] D. Sohler, Z. Dombrádi, J. Timár, O. Sorlin, F. Azaiez, F. Amorini, M. Belleguic, C. Bourgeois, C. Donzaud, J. Duprat, D. Guillemaud-Mueller, F. Ibrahim, J.A. Scarpaci, M. Stanoiu, M.J. Lopez, M.G. Saint-Laurent, F. Becker, F. Sarazin, C. Stodel, G. Voltolini, S.M. Lukyanov, V. Maslov, Y.-E. Penionzhkevich, M. Girod, S. Péru, F. Nowacki, G. Sletten, R. Lucas, C. Theisen, D. Baiborodin, Z. Dlouhy, Physics Letters B 853 (2024) 138694 7 J. Singh, J. Casal, W. Horiuchi et al. J. Mrazek, C. Borcea, A. Bauchet, C.J. Moore, M.J. Taylor, Shape evolution in heavy sulfur isotopes and erosion of the 𝑛 =28 shell closure, Phys. Rev. C 66 (2002) 054302, https://doi .org /10 .1103 /PhysRevC .66 .054302, https://link .aps . org /doi /10 .1103 /PhysRevC .66 .054302. [22] K. Nowak, K. Wimmer, S. Hellgartner, D. Mücher, V. Bildstein, J. Diriken, J. Elseviers, L.P. Gaffney, R. Gernhäuser, J. Iwanicki, J.G. Johansen, M. Huyse, J. Konki, T. Kröll, R. Krücken, R. Lutter, R. Orlandi, J. Pakarinen, R. Raabe, P. Reiter, T. Roger, G. Schrieder, M. Seidlitz, O. Sorlin, P. Van Duppen, N. Warr, H. De Witte, M. Zieli´ nska, Spectroscopy of 46Ar by the (𝑡, 𝑝)two-neutron transfer reaction, Phys. Rev. C 93 (2016) 044335, https://doi .org /10 .1103 /PhysRevC .93 .044335, https:// link .aps .org /doi /10 .1103 /PhysRevC .93 .044335. [23] S. Calinescu, L. Cáceres, S. Grévy, O. Sorlin, Z. Dombrádi, M. Stanoiu, R. Astabatyan, C. Borcea, R. Borcea, M. Bowry, W. Catford, E. Clément, S. Franchoo, R. Garcia, R. Gillibert, I.H. Guerin, I. Kuti, S. Lukyanov, A. Lepailleur, V. Maslov, P. Morfouace, J. Mrazek, F. Negoita, M. Niikura, L. Perrot, Z. Podolyák, C. Petrone, Y. Penionzhkevich, T. Roger, F. Rotaru, D. Sohler, I. Stefan, J.C. Thomas, Z. Vajta, E. Wilson, Coulomb excitation of 44Ca and 46Ar, Phys. Rev. C 93 (2016) 044333, https://doi .org /10 .1103 /PhysRevC .93 .044333, https://link .aps .org /doi / 10 .1103 /PhysRevC .93 .044333. [24] S. Momiyama, K. Wimmer, D. Bazin, J. Belarge, P. Bender, B. Elman, A. Gade, K.W. Kemper, N. Kitamura, B. Longfellow, E. Lunderberg, M. Niikura, S. Ota, P. Schrock, J.A. Tostevin, D. Weisshaar, Shell structure of 43Sand collapse of the 𝑛 =28shell closure, Phys. Rev. C 102 (2020) 034325, https://doi .org /10 .1103 /PhysRevC .102 . 034325, https://link .aps .org /doi /10 .1103 /PhysRevC .102 .034325. [25] B. Longfellow, D. Weisshaar, A. Gade, B.A. Brown, D. Bazin, K.W. Brown, B. Elman, J. Pereira, D. Rhodes, M. Spieker, Shape changes in the 𝑛 =28 island of inversion: collective structures built on configuration-coexisting states in 43S, Phys. Rev. Lett. 125 (2020) 232501, https://doi .org /10 .1103 /PhysRevLett .125 .232501, https://link .aps .org /doi /10 .1103 /PhysRevLett .125 .232501. [26] D.S. Ahn, J. Amano, H. Baba, N. Fukuda, H. Geissel, N. Inabe, S. Ishikawa, N. Iwasa, T. Komatsubara, T. Kubo, K. Kusaka, D.J. Morrissey, T. Nakamura, M. Ohtake, H. Otsu, T. Sakakibara, H. Sato, B.M. Sherrill, Y. Shimizu, T. Sumikama, H. Suzuki, H. Takeda, O.B. Tarasov, H. Ueno, Y. Yanagisawa, K. Yoshida, Discovery of 39Na, Phys. Rev. Lett. 129 (2022) 212502, https://doi .org /10 .1103 /PhysRevLett .129 .212502, https://link .aps .org /doi /10 .1103 /PhysRevLett .129 .212502. [27] K.Y. Zhang, P. Papakonstantinou, M.-H. Mun, Y. Kim, H. Yan, X.-X. Sun, Collapse of the 𝑛 =28 shell closure in the newly discovered 39Na nucleus and the development of deformed halos towards the neutron dripline, Phys. Rev. C 107 (2023) L041303, https://doi .org /10 .1103 /PhysRevC .107 .L041303, https:// link .aps .org /doi /10 .1103 /PhysRevC .107 .L041303. [28] Y. Suzuki, M. Kimura, Triaxial deformation and the disappearance of the 𝑛 =28 shell gap, Phys. Rev. C 104 (2021) 024327, https://doi .org /10 .1103 /PhysRevC .104 . 024327, https://link .aps .org /doi /10 .1103 /PhysRevC .104 .024327. [29] Y. Suzuki, W. Horiuchi, M. Kimura, Erosion of N = 28 shell closure: shape coexistence and monopole transition, Prog. Theor. Exp. Phys. 2022 (6) (2022) 063D02, https://doi .org /10 .1093 /ptep /ptac071, https://academic .oup .com /ptep / article -pdf /2022 /6 /063D02 /44062687 /ptac071 .pdf. [30] J. Singh, J. Casal, W. Horiuchi, L. Fortunato, A. Vitturi, Exploring two-neutron halo formation in the ground state of 29Fwithin a three-body model, Phys. Rev. C 101 (2020) 024310, https://doi .org /10 .1103 /PhysRevC .101 .024310, https:// link .aps .org /doi /10 .1103 /PhysRevC .101 .024310. [31] L. Fortunato, J. Casal, W. Horiuchi, J. Singh, A. Vitturi, The 29f nucleus as a lighthouse on the coast of the island of inversion, Commun. Phys. 3(1) (2020) 132–136, https://doi .org /10 .1038 /s42005 -020 -00402 -5. [32] J. Casal, J. Singh, L. Fortunato, W. Horiuchi, A. Vitturi, Electric dipole response of low-lying excitations in the two-neutron halo nucleus 29F, Phys. Rev. C 102 (2020) 064627, https://doi .org /10 .1103 /PhysRevC .102 .064627, https://link .aps .org /doi / 10 .1103 /PhysRevC .102 .064627. [33] F. Nowacki, A. Obertelli, A. Poves, The neutron-rich edge of the nuclear landscape: experiment and theory, Prog. Part. Nucl. Phys. 120 (2021) 103866, https://doi .org / 10 .1016 /j .ppnp .2021 .103866, https://www .sciencedirect .com /science /article /pii / S014664102100020X. [34] M. Takechi, S. Suzuki, D. Nishimura, M. Fukuda, T. Ohtsubo, M. Nagashima, T. Suzuki, T. Yamaguchi, A. Ozawa, T. Moriguchi, H. Ohishi, T. Sumikama, H. Geissel, N. Aoi, R.-J. Chen, D.-Q. Fang, N. Fukuda, S. Fukuoka, H. Furuki, N. Inabe, Y. Ishibashi, T. Itoh, T. Izumikawa, D. Kameda, T. Kubo, M. Lantz, C.S. Lee, Y.- G. Ma, K. Matsuta, M. Mihara, S. Momota, D. Nagae, R. Nishikiori, T. Niwa, T. Ohnishi, K. Okumura, M. Ohtake, T. Ogura, H. Sakurai, K. Sato, Y. Shimbara, H. Suzuki, H. Takeda, S. Takeuchi, K. Tanaka, M. Tanaka, H. Uenishi, M. Winkler, Y. Yanagisawa, S. Watanabe, K. Minomo, S. Tagami, M. Shimada, M. Kimura, T. Matsumoto, Y.R. Shimizu, M. Yahiro, Evidence of halo structure in 37Mg observed via reaction cross sections and intruder orbitals beyond the island of inversion, Phys. Rev. C 90 (2014) 061305(R), https://doi .org /10 .1103 /PhysRevC .90 .061305, https://link .aps .org /doi /10 .1103 /PhysRevC .90 .061305. [35] N. Kobayashi, T. Nakamura, Y. Kondo, J.A. Tostevin, Y. Utsuno, N. Aoi, H. Baba, R. Barthelemy, M.A. Famiano, N. Fukuda, N. Inabe, M. Ishihara, R. Kanungo, S. Kim, T. Kubo, G.S. Lee, H.S. Lee, M. Matsushita, T. Motobayashi, T. Ohnishi, N.A. Orr, H. Otsu, T. Otsuka, T. Sako, H. Sakurai, Y. Satou, T. Sumikama, H. Takeda, S. Takeuchi, R. Tanaka, Y. Togano, K. Yoneda, Observation of a 𝑝-wave one-neutron halo configuration in 37Mg, Phys. Rev. Lett. 112 (2014) 242501, https://doi .org / 10 .1103 /PhysRevLett .112 .242501, https://link .aps .org /doi /10 .1103 /PhysRevLett . 112 .242501. [36] Neelam Shubhchintak, R. Chatterjee, R. Shyam, K. Tsushima, Coulomb breakup of 37mg and its ground state structure, Nucl. Phys. A 939 (2015) 101–120, https://doi .org /10 .1016 /j .nuclphysa .2015 .03 .011, https://www . sciencedirect .com /science /article /pii /S0375947415000834. [37] K. Zhang, S. Yang, J. An, S. Zhang, P. Papakonstantinou, M.-H. Mun, Y. Kim, H. Yan, Missed prediction of the neutron halo in 37mg, Phys. Lett. B 844 (2023) 138112, https://doi .org /10 .1016 /j .physletb .2023 .138112, https://www .sciencedirect .com / science /article /pii /S037026932300446X. [38] J.-L. An, K.-Y. Zhang, Q. Lu, S.-Y. Zhong, S.-S. Zhang, A unified description of the halo nucleus 37mg from microscopic structure to reaction observables, Phys. Lett. B 849 (2024) 138422, https://doi .org /10 .1016 /j .physletb .2023 .138422, https:// www .sciencedirect .com /science /article /pii /S0370269323007554. [39] I. Hamamoto, Nilsson diagrams for light neutron-rich nuclei with weakly-bound neutrons, Phys. Rev. C 76 (2007) 054319, https://doi .org /10 .1103 /PhysRevC .76 . 054319, https://link .aps .org /doi /10 .1103 /PhysRevC .76 .054319. [40] H. Nakada, K. Takayama, Intertwined effects of pairing and deformation on neutron halos in magnesium isotopes, Phys. Rev. C 98 (2018) 011301, https://doi .org /10 .1103 /PhysRevC .98 .011301, https://link .aps .org /doi /10 .1103 / PhysRevC .98 .011301. [41] A.O. Macchiavelli, H.L. Crawford, P. Fallon, R.M. Clark, A. Poves, Weak binding effects on the structure of 40mg, Eur. Phys. J. A 58 (2022) 66, https://doi .org /10 . 1140 /epja /s10050 -022 -00719 -5. [42] M. Wang, W. Huang, F. Kondev, G. Audi, S. Naimi, The AME 2020 atomic mass evaluation (II). Tables, graphs and references*, Chin. Phys. C 45 (3) (2021) 030003, https://doi .org /10 .1088 /1674 -1137 /abddaf. [43] T. Baumann, A.M. Amthor, D. Bazin, B.A. Brown, C.M. Folden III, A. Gade, T.N. Ginter, M. Hausmann, M. Matoš, D.J. Morrissey, M. Portillo, A. Schiller, B.M. Sherrill, A. Stolz, O.B. Tarasov, M. Thoennessen, Discovery of 40mg and 42al suggests neutron drip-line slant towards heavier isotopes, Nature 449 (2007) 1022, https:// doi .org /10 .1038 /nature06213. [44] Q. Yuan, J. Li, H. Li, Ab initio calculations for well deformed nuclei: 40mg and 42si, Phys. Lett. B 848 (2024) 138331, https://doi .org /10 .1016 /j .physletb .2023 .138331, https://www .sciencedirect .com /science /article /pii /S0370269323006652. [45] L. Fortunato, R. Chatterjee, J. Singh, A. Vitturi, Pairing in the continuum: the quadrupole response of the Borromean nucleus 6He, Phys. Rev. C 90 (2014) 064301, https://doi .org /10 .1103 /PhysRevC .90 .064301, https://link .aps .org /doi / 10 .1103 /PhysRevC .90 .064301. [46] J. Singh, L. Fortunato, A. Vitturi, R. Chatterjee, Electric multipole response of the halo nucleus 6he, Eur. Phys. J. A 52 (7) (2016) 209, https://doi .org /10 .1140 /epja / i2016 -16209 -8. [47] I. Tanihata, H. Savajols, R. Kanungo, Recent experimental progress in nuclear halo structure studies, Prog. Part. Nucl. Phys. 68 (2013) 215–313, https://doi . org /10 .1016 /j .ppnp .2012 .07 .001, http://www .sciencedirect .com /science /article / pii /S0146641012001081. [48] M. Zhukov, B. Danilin, D. Fedorov, J. Bang, I. Thompson, J. Vaagen, Bound state properties of borromean halo nuclei: 6he and 11li, Phys. Rep. 231 (4) (1993) 151–199, https://doi .org /10 .1016 /0370 -1573(93 )90141 -Y, http://www . sciencedirect .com /science /article /pii /037015739390141Y. [49] E. Nielsen, D. Fedorov, A. Jensen, E. Garrido, The three-body problem with short-range interactions, Phys. Rep. 347 (5) (2001) 373–459, https://doi .org /10 . 1016 /S0370 -1573(00 )00107 -1, http://www .sciencedirect .com /science /article /pii / S0370157300001071. [50] J. Casal, Two-nucleon emitters within a pseudostate method: the case of 6Be and 16Be, Phys. Rev. C 97 (2018) 034613, https://doi .org /10 .1103 /PhysRevC .97 . 034613, https://link .aps .org /doi /10 .1103 /PhysRevC .97 .034613. [51] P. Descouvemont, C. Daniel, D. Baye, Three-body systems with Lagrange-mesh techniques in hyperspherical coordinates, Phys. Rev. C 67 (2003) 044309, https://doi .org /10 .1103 /PhysRevC .67 .044309, https://link .aps .org /doi /10 .1103 / PhysRevC .67 .044309. [52] T. Matsumoto, E. Hiyama, M. Yahiro, K. Ogata, Y. Iseri, M. Kamimura, Nucl. Phys. A 738 (2004) 471, https://doi .org /10 .1016 /j .nuclphysa .2004 .04 .089. [53] M. Rodríguez-Gallardo, J.M. Arias, J. Gómez-Camacho, A.M. Moro, I.J. Thompson, J.A. Tostevin, Three-body continuum discretization in a basis of transformed harmonic oscillator states, Phys. Rev. C 72 (2005) 024007, https://doi .org /10 .1103 / PhysRevC .72 .024007, https://link .aps .org /doi /10 .1103 /PhysRevC .72 .024007. [54] J. Casal, M. Rodríguez-Gallardo, J.M. Arias, Analytical transformed harmonic oscillator basis for three-body nuclei of astrophysical interest: application to 6he, Phys. Rev. C 88 (2013) 014327, https://doi .org /10 .1103 /PhysRevC .88 .014327, https:// link .aps .org /doi /10 .1103 /PhysRevC .88 .014327. [55] J. Casal, Weakly-bound three-body nuclear systems: Structure, reactions and astrophysical implications, Ph.D. thesis, Universidad de Sevilla, 2016, https://idus .us .es / xmlui /handle /11441 /41814. [56] I. Thompson, F. Nunes, B. Danilin, Face: a tool for three body Faddeev calculations with core excitation, Comput. Phys. Commun. 161 (1) (2004) 87–107, https:// doi .org /10 .1016 /j .cpc .2004 .03 .007, http://www .sciencedirect .com /science / article /pii /S0010465504002140. [57] R. de Diego, E. Garrido, D.V. Fedorov, A.S. Jensen, Relative production rates of 6 he, 9 be, 12 c in astrophysical environments, Europhys. Lett. 90 (5) (2010) 52001, http://stacks .iop .org /0295 -5075 /90 /i =5 /a =52001. Physics Letters B 853 (2024) 138694 8 J. Singh, J. Casal, W. Horiuchi et al. [58] G. Singh, J. Singh, J. Casal, L. Fortunato, Exploring the halo character and dipole response in the dripline nucleus 31F, Phys. Rev. C 105 (2022) 014328, https://doi .org /10 .1103 /PhysRevC .105 .014328, https://link .aps .org /doi /10 .1103 / PhysRevC .105 .014328. [59] K. Fossez, J. Rotureau, N. Michel, Q. Liu, W. Nazarewicz, Single-particle and collective motion in unbound deformed 39Mg, Phys. Rev. C 94 (2016) 054302, https://doi .org /10 .1103 /PhysRevC .94 .054302, https://link .aps .org /doi / 10 .1103 /PhysRevC .94 .054302. [60] W. Horiuchi, Y. Suzuki, P. Capel, D. Baye, Probing the weakly-bound neutron orbit of 31Ne with total reaction and one-neutron removal cross sections, Phys. Rev. C 81 (2010) 024606, https://doi .org /10 .1103 /PhysRevC .81 .024606, https://link .aps . org /doi /10 .1103 /PhysRevC .81 .024606. [61] A. Bohr, B.R. Mottelson, Nuclear Structure, Benjamin, New York, 1969. [62] D. Baye, Supersymmetry between deep and shallow nucleus-nucleus potentials, Phys. Rev. Lett. 58 (1987) 2738–2741, https://doi .org /10 .1103 /PhysRevLett .58 . 2738, https://link .aps .org /doi /10 .1103 /PhysRevLett .58 .2738. [63] D. Baye, Phase-equivalent potentials from supersymmetry, J. Phys. A, Math. Gen. 20 (16) (1987) 5529–5540, https://doi .org /10 .1088 /0305 -4470 /20 /16 /027. [64] K. Hagino, H. Sagawa, Decay dynamics of the unbound 25Oand 26Onuclei, Phys. Rev. C 93 (2016) 034330, https://doi .org /10 .1103 /PhysRevC .93 .034330, https:// link .aps .org /doi /10 .1103 /PhysRevC .93 .034330. [65] T. Otsuka, T. Suzuki, J.D. Holt, A. Schwenk, Y. Akaishi, Three-body forces and the limit of oxygen isotopes, Phys. Rev. Lett. 105 (2010) 032501, https://doi .org / 10 .1103 /PhysRevLett .105 .032501, https://link .aps .org /doi /10 .1103 /PhysRevLett . 105 .032501. [66] J. Casal, E. Garrido, Three-body structure of 19B: finite-range effects in two-neutron halo nuclei, Phys. Rev. C 102 (2020) 051304, https://doi .org /10 .1103 /PhysRevC . 102 .051304, https://link .aps .org /doi /10 .1103 /PhysRevC .102 .051304. [67] D. Gogny, P. Pires, R.D. Tourreil, A smooth realistic local nucleon-nucleon force suitable for nuclear Hartree-Fock calculations, Phys. Lett. B 32 (7) (1970) 591–595, https://doi .org /10 .1016 /0370 -2693(70 )90552 -6, http://www . sciencedirect .com /science /article /pii /0370269370905526. [68] A.E. Lovell, F.M. Nunes, I.J. Thompson, Three-body model for the two-neutron emission of 16Be, Phys. Rev. C 95 (2017) 034605, https://doi .org /10 .1103 /PhysRevC . 95 .034605, https://link .aps .org /doi /10 .1103 /PhysRevC .95 .034605. [69] S. Watanabe, K. Minomo, M. Shimada, S. Tagami, M. Kimura, M. Takechi, M. Fukuda, D. Nishimura, T. Suzuki, T. Matsumoto, Y.R. Shimizu, M. Yahiro, Ground-state properties of neutron-rich mg isotopes, Phys. Rev. C 89 (2014) 044610, https://doi .org /10 .1103 /PhysRevC .89 .044610, https://link .aps .org /doi / 10 .1103 /PhysRevC .89 .044610. [70] L. Geng, H. Toki, A. Ozawa, J. Meng, Proton and neutron skins of light nuclei within the relativistic mean field theory, Nucl. Phys. A 730 (1) (2004) 80–94, https://doi .org /10 .1016 /j .nuclphysa .2003 .10 .014, https://www . sciencedirect .com /science /article /pii /S0375947403017998. [71] H. Masui, W. Horiuchi, M. Kimura, Two-neutron halo structure of 31Fand a novel pairing antihalo effect, Phys. Rev. C 101 (2020) 041303, https://doi .org /10 .1103 / PhysRevC .101 .041303, https://link .aps .org /doi /10 .1103 /PhysRevC .101 .041303. [72] A. Ozawa, O. Bochkarev, L. Chulkov, D. Cortina, H. Geissel, M. Hellström, M. Ivanov, R. Janik, K. Kimura, T. Kobayashi, A. Korsheninnikov, G. Münzenberg, F. Nickel, Y. Ogawa, A. Ogloblin, M. Pfützner, V. Pribora, H. Simon, B. Sitár, P. Strmen, K. Sümmerer, T. Suzuki, I. Tanihata, M. Winkler, K. Yoshida, Measurements of interaction cross sections for light neutron-rich nuclei at relativistic energies and determination of effective matter radii, Nucl. Phys. A 691 (3) (2001) 599–617, https:// doi .org /10 .1016 /S0375 -9474(01 )00563 -2, http://www .sciencedirect .com /science / article /pii /S0375947401005632. [73] R. Kanungo, A. Prochazka, W. Horiuchi, C. Nociforo, T. Aumann, D. Boutin, D. Cortina-Gil, B. Davids, M. Diakaki, F. Farinon, H. Geissel, R. Gernhäuser, J. Gerl, R. Janik, B. Jonson, B. Kindler, R. Knöbel, R. Krücken, M. Lantz, H. Lenske, Y. Litvinov, B. Lommel, K. Mahata, P. Maierbeck, A. Musumarra, T. Nilsson, C. Perro, C. Scheidenberger, B. Sitar, P. Strmen, B. Sun, Y. Suzuki, I. Szarka, I. Tanihata, Y. Utsuno, H. Weick, M. Winkler, Matter radii of 32−35mg, Phys. Rev. C 83 (2011) 021302, https://doi .org /10 .1103 /PhysRevC .83 .021302, https:// link .aps .org /doi /10 .1103 /PhysRevC .83 .021302. [74] R.J. Glauber, High-energy collision theory, in: W.E. Brittin, L.G. Dunham (Eds.), Lectures in Theoretical Physics, vol. 1, Interscience, New York, 1959, p. 315. [75] B. Abu-Ibrahim, Y. Suzuki, Utility of nucleon-target profile function in cross section calculations, Phys. Rev. C 61 (2000) 051601, https://doi .org /10 .1103 /PhysRevC . 61 .051601, https://link .aps .org /doi /10 .1103 /PhysRevC .61 .051601. [76] B. Abu-Ibrahim, W. Horiuchi, A. Kohama, Y. Suzuki, Reaction cross sections of carbon isotopes incident on a proton, Phys. Rev. C 77 (2008) 034607, https://doi .org /10 .1103 /PhysRevC .77 .034607, https://link .aps .org /doi /10 .1103 / PhysRevC .77 .034607. [77] B. Abu-Ibrahim, W. Horiuchi, A. Kohama, Y. Suzuki, Erratum: reaction cross sections of carbon isotopes incident on a proton, Phys. Rev. C 77 (2008) 034607, Phys. Rev. C 80 (2009) 029903, https://doi .org /10 .1103 /PhysRevC .80 .029903, https:// link .aps .org /doi /10 .1103 /PhysRevC .80 .029903. [78] B. Abu-Ibrahim, W. Horiuchi, A. Kohama, Y. Suzuki, Publisher’s note: reaction cross sections of carbon isotopes incident on a proton, Phys. Rev. C 77 (2008) 034607, Phys. Rev. C 81 (2010) 019901, https://doi .org /10 .1103 /PhysRevC .81 .019901, https://link .aps .org /doi /10 .1103 /PhysRevC .81 .019901. [79] W. Horiuchi, Y. Suzuki, 22C: an 𝑠-wave two-neutron halo nucleus, Phys. Rev. C 74 (2006) 034311, https://doi .org /10 .1103 /PhysRevC .74 .034311, https://link .aps . org /doi /10 .1103 /PhysRevC .74 .034311. [80] W. Horiuchi, Y. Suzuki, B. Abu-Ibrahim, A. Kohama, Systematic analysis of reaction cross sections of carbon isotopes, Phys. Rev. C 75 (2007) 044607, https://doi .org /10 .1103 /PhysRevC .75 .044607, https://link .aps .org /doi /10 .1103 / PhysRevC .75 .044607. [81] W. Horiuchi, Y. Suzuki, B. Abu-Ibrahim, A. Kohama, Erratum: systematic analysis of reaction cross sections of carbon isotopes, Phys. Rev. C 75 (2007) 044607, Phys. Rev. C 76 (2007) 039903, https://doi .org /10 .1103 /PhysRevC .76 .039903, https:// link .aps .org /doi /10 .1103 /PhysRevC .76 .039903. [82] W. Horiuchi, T. Inakura, T. Nakatsukasa, Y. Suzuki, Glauber-model analysis of total reaction cross sections for ne, mg, si, and s isotopes with Skyrme-Hartree-Fock densities, Phys. Rev. C 86 (2012) 024614, https://doi .org /10 .1103 /PhysRevC .86 . 024614, https://link .aps .org /doi /10 .1103 /PhysRevC .86 .024614. [83] W. Horiuchi, T. Inakura, T. Nakatsukasa, Y. Suzuki, Systematic analysis of total reaction cross sections of unstable nuclei with Glauber theory, JPS Conf. Proc. 6 (2015) 030079, https://doi .org /10 .7566 /JPSCP .6 .030079, https://journals .jps .jp / doi /abs /10 .7566 /JPSCP .6 .030079. [84] T. Nagahisa, W. Horiuchi, Examination of the 22Cradius determination with interaction cross sections, Phys. Rev. C 97 (2018) 054614, https://doi .org /10 .1103 / PhysRevC .97 .054614, https://link .aps .org /doi /10 .1103 /PhysRevC .97 .054614.