Evolution of the N = 28 shell closure: a test bench for nuclear forces

As the Ar isotopes have an open proton shell configuration, the 2⁺₁ state is likely to be mainly of proton origin there. The 2⁺₁ energy at the N = 20 and 28 shell closures is smaller in the Ar isotopes as it comes naturally from the proton re-coupling inside the sd shells. However, a rise in 2⁺ energy is found, witnessing the presence of neutron shell closures. It is worth noting that two very different values of B(E2;0⁺₁ → 2⁺₁) have been determined in ⁴⁶Ar (figure 3). They arise from two different experimental techniques. On the one hand, two experiments were carried at the NSCL facility to study the Coulomb excitation of ⁴⁶Ar. They led to consistent results, with a rather small B(E2)_up value of 218(31) e²fm⁴ [13]. On the other hand, a fairly large B(E2)_up value of 570 ⁺³³⁵₋₁₆₀ e²fm⁴ was deduced from another experiment performed at the Legnaro facility by Mengoni et al [14]. In that work, ⁴⁶Ar was produced by means of multi-nucleon transfer reaction. The lifetime of the 2⁺₁ state was determined by using the differential recoil distance Doppler shift method. In addition, a tentative 4⁺₁ state has been proposed at 3892 keV by Dombrádi et al [17], leading to E(4⁺₁/2⁺₁) = 2.47, a value which lies between the limits of the vibrator and rotor nuclei.

Comparison of these two B(E2) values to models is very instructive. The recent relativistic Hartree–Bogoliubov model based on the DD-PC1 energy density functional [18] reproduces very well the low value of [13], while the shell-model (SM) calculations of Nowacki and Poves [19] agree with the large value of [14]. It therefore seems that although only two protons below the doubly magic ⁴⁸Ca nucleus, ⁴⁶Ar is still a challenging nucleus to modelize.

Interesting to add is the fact that the B(E2) in ⁴⁷Ar [20] is close to the lowest value measured in ⁴⁶Ar, i.e. at variance with the SM predictions, while the B(E2) experimental result and SM calculations agree with an enhanced collectivity for ⁴⁸Ar [20]. SM calculations also account for the spectroscopy of higher energy states in ⁴⁸Ar obtained from deep-inelastic transfer reactions, which show that its deformation is likely non-axially symmetric [15]. To conclude, a new measurement of the B(E2) in ⁴⁶Ar is therefore called for to see if a local discrepancy between SM calculations and experimental results persists at N = 28.

The B(E2;0⁺₁ → 2⁺₁) and 2⁺₁ energy values in ⁴⁴S, determined using Coulomb excitation at high energy at the NSCL facility by Glasmacher et al ([21] and references therein), point to a configuration which is intermediate between a spherical and a deformed nucleus. Evidence for shape coexistence was found at GANIL by determining the decay rates of the isomeric 0⁺₂ state at 1365(1) keV to the 2⁺₁ at 1329(1) keV and 0⁺₁ ground state through delayed electron and γ spectroscopy [22]. Comparisons to SM calculations point to prolate-spherical shape coexistence. A schematic two-level mixing model was used to extract a weak mixing between the two configurations, the 0⁺₁ state having a deformed configuration (i.e. a two-neutron particle–hole 2p2h excitation across the N = 28 spherical gap) and the 0⁺₂ state, a spherical configuration (i.e. a zero-neutron particle–hole excitation 0p0h). New excited states in ⁴⁴S were found afterwards [23, 24], which lend support to the shape coexistence hypothesis. By using the two-proton knockout reaction from ⁴⁶Ar at intermediate beam energy at NSCL, four new excited states were observed [23]. The authors proposed that one of them has a strongly deformed configuration, due to the promotion of a single neutron (1p1h) across the N = 28 gap. It would follow that three configurations would coexist in ⁴⁴S, corresponding to zero-, one- and two-neutron particle–hole excitations. The one-proton knockout reaction from ⁴⁵Cl gave access to other excited states and their γ decays were measured using BaF₂ detectors at GANIL [24]. In particular, the existence of the 2⁺₂ state at 2156(49) keV is confirmed and is likely the expected 'spherical' 2⁺ state. The Hartree–Bogoliubov model based on the DD-PC1 energy density functional [18] gives a relatively good agreement with experimental results, with the exception of a too large mixing between the 0⁺₁ and 0⁺₂ states. The Gogny force with the D1S set of parameters also gives a good description of ⁴⁴S, as described in [25, 26]. Shape coexistence has also been evidenced in the neighboring ⁴³S₂₇ nucleus, which will be described in section 4.

As to the ⁴²Si₂₈ nucleus, Bastin et al [9] have established a 2⁺ state at 770(19) keV. This experiment used nucleon removal reactions from secondary beams centered around ⁴⁴S at intermediate energy to produce the ⁴²Si nuclei and study the γ-rays from their de-excitation in flight. The detection of the γ-rays was achieved by arrays of detectors which surrounded the production target in which the reactions occurred. The dramatic decrease of the 2⁺₁ energy in ⁴²Si is a proof of the disappearance of the spherical N = 28 shell closure at Z = 14. This extremely low energy of 770 keV—actually one of the smallest among nuclei having a similar atomic mass—cannot be obtained solely from neutron excitations. Proton-core excitations should play an important role, which could in principle be evidenced by measuring the evolution of the B(E2) values in the Si isotopic chain while reaching N = 28. The B(E2) values in the ₁₄Si isotopic chain [27] seem to rise after N = 20, but not as much as in the ₁₆S one (see the right part of figure 3). Whether the B(E2) values remain small, steadily increase up to N = 28 or follow a parabola cannot be judged at the present time as the quoted experimental error bars are too large. A reduced Z = 14 shell gap would dramatically increase the B(E2) values, as protons are carrying most of the effective charge in the nucleus. The sole decrease of the N = 28 gap would barely change the 2⁺₁ and B(E2) values [9]. The effect of the tensor force is to reduce the 2⁺₁ energy and enhance the B(E2) value at N = 28 as shown in figure 4, leading to B(E2;0⁺₁ → 2⁺₁) ∼ 430 e²fm⁴ in ⁴²Si. Without implementing tensor parts in the monopole terms, the B(E2) in ⁴²Si drops down to 150 e²fm⁴. One could deduce that the studies of the 2⁺₁ state and the B(E2) value in ⁴²Si are essential to ascertain the role of tensor forces at N = 28.

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Most of the MF approaches do not include a tensor force. Indeed, the B(E2) value of ⁴²Si has been calculated in the relativistic Hartree–Bogoliubov model of Li et al [18] and in the Gogny (D1S) model of Rodríguez-Guzmán et al [26], which do not contain tensor force. However, their B(E2) values differ significantly: 200 e²fm⁴ in [18] and about 470 e²fm⁴ in [26]. As far as the 2⁺₁ energy is concerned, the value of [18] is twice as large as the experimental value, while that of [26] is closer to experiment. Therefore the B(E2) values should be measured and compared to theory in the whole Si isotopic chain to see if a significant increase is occurring at N = 28. The evolution of the sd proton orbits should be used as well, because they influence strongly the B(E2) and E(2⁺) values.

Further discussions on the implementation of tensor interaction and its role in the evolution of the gaps could be found, for instance, in [29–32].

The results of a new experimental study of the excited states of ⁴²Si have just been published [33]. Thanks to the intense radioactive beams provided at RIKEN RIBF which enable γ–γ coincidence measurements, the most probable candidate for the transition from the yrast 4⁺ state to the 2⁺ was identified, leading to a 4⁺₁ energy of 2173(14) keV. Then the energy ratio, R_4/2 ∼ 2.9, corresponds to a well-deformed rotor. In addition, two other γ lines were measured at high energy (at 2032(9) and 2357(15) keV), which deserve to be better characterized in order to assign the other excited states of ⁴²Si.

Some years ago, the observation of three ⁴⁰Mg nuclei in the fragmentation of a primary beam of ⁴⁸Ca impinging on a W target ended speculations about the location of the neutron drip line at Z = 12 [34]. This isotope is predicted to lie inside the neutron drip line in many theoretical calculations (see, for instance, [35] and references therein). The relativistic Hartree–Bogoliubov model for triaxial nuclei was used to calculate the binding energy map of ⁴⁰Mg in the β–γ-plane [18], which predicts that this extremely neutron-rich isotope shows a deep prolate minimum at (β,γ) = (0.45,0°). SM [19] as well as Gogny (D1S) [26] calculations predict an extremely prolate rotor as well. Identification of the first excited state of ⁴⁰Mg remains an ambitious challenge for the future. Conjectures about shell evolutions below ⁴²Si will be provided in section 6.5.

In this section, we investigate the role of neutron–neutron interactions in creating the neutron SO shell gaps. The three-body forces are needed to create these gaps, which would not be obtained if realistic two-body forces would be used exclusively. We start with the study of the N = 28 shell gap and generalize this study to other SO shell gaps as N = 14 and 50. We conclude that the three-body forces seem to play a similar and essential role in creating the major shell gaps that lead to SO magic nuclei. We propose an empirical rule to predict the evolution of high-j orbits in superheavy nuclei as well as to create a sub-shell gap at N = 90, which could be of importance for the r-process nucleosynthesis.

Search for fp single particles has been carried out using transfer reactions [58, 59] at N = 20 and 28 by studying the particle and hole strengths around the ⁴⁰Ca and ⁴⁸Ca nuclei, respectively. These experiments show that the N = 28 shell gap strongly increases when adding neutrons into the f_7/2 orbit, as the neutron f_7/2 (p_3/2) orbit gains (loses) binding energy. The gain in energy of the N = 28 gap from N = 20 to 28 is about δGⁿⁿ(28) = 3 MeV (see the middle of figure 10).

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The mechanism which creates the large N = 28 shell gap, between the f_7/2 and p_3/2 orbits, is probably due to a strongly attractive Vⁿⁿ_f7/2 f7/2 and a repulsive Vⁿⁿ_f7/2 p3/2 monopole term⁶. The increase of the N = 28 shell gap, δGⁿⁿ(28), due to the neutron–neutron interactions can be written as a function of x (the number of neutrons added in the f_7/2 orbit) as

$$\begin{equation} \delta G^{\mathrm{nn}}(28)= x \left[V^{\mathrm{nn}}_{{\mathrm{f}}_{7/2}{\mathrm{p}}_{3/2}}\right] - (x-1) \left[V^{\mathrm{nn}}_{{\mathrm{f}}_{7/2}{\mathrm{f}}_{7/2}}\right]. \end{equation} \tag{ 8 }$$

We note that the (x − 1) term applies when neutrons occupy the same shell.

Effective two-body interactions derived from realistic interactions could not account for this increase of the N = 28 gap. For instance, it has recently been shown by Holt et al [60] that the N = 28 gap is almost constant using two-body forces and grows when three-body forces are taken into account. A more recent work which uses interactions from chiral effective field theory leads to the same conclusion: the three-nucleon forces create an N = 28 magic number [61] and generate the shell closure in ⁴⁸Ca. More generally, it was proposed by Zuker [62] that the implementation of a three-body force can solve several deficiencies in nuclear structure.

The N = 14 subshell gap, which is located between the d_5/2 and s_1/2 orbits, also comes from the SO splitting and shares similar properties as the N = 28 one. As shown in figure 10, it grows by about δGⁿⁿ(14) = 2.7 MeV from ¹⁷O to ²³O as the neutron d_5/2 orbit is filled. Similar to the N = 28 gap, equation (8) applies and the large value of the N = 14 gap at N = 14 is due to the strongly attractive (repulsive) Vⁿⁿ_d5/2 d5/2 (Vⁿⁿ_d5/2 s1/2) monopole terms.

From a phenomenological point of view, we can reasonably expect that a similar increase in binding energy of major shells having ℓ and s values aligned ℓ↑, i.e. d_5/2, f_7/2, g_9/2, h_11/2, will be occurring during their filling. This generally leads to an increase of shell gap by

$$\begin{equation} \delta G^{\mathrm{nn}}(\mathrm{SO})= x \left[V^{\mathrm{nn}}_{\ell _\uparrow (\ell-2)_\uparrow}\right] - (x-1) \left[V^{\mathrm{nn}}_{\ell _\uparrow \ell _\uparrow}\right]. \end{equation} \tag{ 9 }$$

In particular, we could look at what is expected for the N = 50 shell gap from ⁶⁸Ni to ⁷⁸Ni, where fewer experimental data are available so far. Based on equation (9) the gap formed between the g_9/2 and d_5/2 orbits is expected to grow as the 1g_9/2 orbit is filled. The relevant effective monopole terms (Vⁿⁿ_g9/2 g9/2 ≃ − 200 keV and Vⁿⁿ_g9/2 d5/2 ≃ + 130 keV) can be extracted from the spectroscopy of ^88,90Zr (see figure 6 of [3]). When renormalizing the monopole terms to A ≃ 75 (using the hypothesis that V scales with A^−1/3) we find that the N = 50 gap should increase by about 3.1 MeV between N = 40 and 50. The right-hand side of figure 10 displays the tentative change of the N = 50 gap in the Ni isotopic chain. The S_n value of ⁶⁹Ni as well as the preliminary energy of the 2d_5/2 state at ≃2.6 MeV derived from [63] are used. Note that a correction in binding energy is applied to S_n(⁶⁹Ni). Indeed, it is likely that the ⁶⁸Ni₄₀ ground state has a mixed contribution with the 0⁺₂ state at 1.77 MeV. Taking a mixing of 50%(νg_9/2)² + 50%(νp_1/2)²,⁷ it implies that before mixing the (p_1/2)² configuration of ⁶⁸Ni is likely less bound by about 0.9 MeV. Thus starting from 2.6 MeV in ⁶⁹Ni, the N = 50 shell gap would reach (2.6 + 3.1 =) 5.8 MeV in ⁷⁸Ni. This is somehow larger than what is extrapolated from the N = 50 trend of binding energies [64] and with the SM predictions of [65]. Note that a small change in one of the Vⁿⁿ values given above can drastically modify the value of the increased gap, as ten neutrons (x =10) are involved here between N = 40 and 50 (see equation (9)).

If this later increase of shell gap is confirmed, this will remarkably show that similar forces (including the three-body term) are at play in the various regions of the chart of nuclides to produce such large SO shell gaps. Even more a priori surprising is the increase of all the N = 14, 28 and 50 shell gaps by a similar value of about 3 MeV. This comes from the fact that two competing terms are involved in enlarging the gap in equation (9). The two monopoles decrease with A^−1/3 as the nucleus grows in size, but they are multiplied by the number of neutrons x in the ℓ↑ orbit, which increases between N = 14 (x = 6 in d_5/2), N = 28 (x = 8 in f_7/2) and N = 50 (x = 10 in g_9/2). We could tentatively propose as an empirical rule that, during the filling of the ℓ↑ orbit, the variation of the major shell gaps of SO origin amounts to about 3 MeV. Although rather crude, this empirical rule could be used to predict the evolution of the (j↑,(j − 2)↑) orbits as a function of the filling of the j↑ orbit. However, the pairing effect will dilute the orbital occupancy in heavier systems, as mentioned below for the A = 132 region of mass.

As for the next shell N = 82, we probably expect the same increase of the h_11/2–f_7/2 spacing while filling the h_11/2 orbit. However, this increase of gap could hardly be evidenced experimentally as the h_11/2 orbit is located close to two low-ℓ orbits, s_1/2 and d_3/2. So the pairing correlations dilute the occupancy of the high-j orbit with the two others, implying that the h_11/2 orbit is gradually filling when adding 18 neutrons from N = 65 to 82. Consequently, the corresponding evolution of the h_11/2 binding energy is not only governed by the attractive Vⁿⁿ_h11/2 h11/2 interaction, but also by the repulsive ones, Vⁿⁿ_h11/2 s1/2 and Vⁿⁿ_h11/2 d3/2. Therefore, in this case, the SO empirical rule could be used as a guidance to constrain the spacing between the ℓ↑ and (ℓ − 2)↑ orbits at the MF level, before taking the role of correlations into account. It is important to point out that none of the self-consistent calculations reproduce the experimental location of the νh_11/2 orbit in the doubly magic ¹³²Sn, i.e. when this orbit is completely filled. All MF models using the Skyrme and Gogny forces or relativistic NL3 and NL-Z2 forces predict that this high-j orbit is located just below the N = 82 gap, the νd_3/2 orbit being more bound by about 1 MeV (see figure 7 of [66]), while the ground state of ¹³¹Sn has spin 3/2⁺ and its first excited state at 65 keV has spin 11/2⁻. One surmises that the lack of three-body forces is responsible for the discrepancy. Indeed, these forces would strongly enhance the binding energy of the filled νh_11/2 orbit.

An interesting effect of the three-body forces would be the creation of a gap at ¹⁴⁰Sn₉₀, between the 2f_7/2 and 2p_3/2 shells, as proposed by Sarkar and Saha Sarkar [67]. Indeed, the neutron–neutron monopole terms (and possibly the three-body terms) which intervene here are somehow similar to the ones at N = 28. The only modifications are that the nodes of the wave functions differ, and that the monopole term for A ≃ 48 has to be downscaled to account for the reduction of the interaction energy as the nucleus grows in size at A ≃ 140. When using equation (8), the suitable downscaled monopole terms, and the known 2f_7/2 − 2p_3/2 spacing of 0.85 MeV in ¹³³Sn [68, 69] it is found that the N = 90 gap is expected to grow by about 2.2 MeV while filling the 2f_7/2 orbit. This leads to a subshell gap of about 3.1 MeV at ¹⁴⁰Sn. If established, this subshell gap would bring an additional credit to the mechanism of shell gap creation by three-body forces in another region of the chart of nuclides. In addition, this subshell could bring a local waiting point around ¹⁴⁰Sn in the r-process nucleosynthesis.

In summary, the three-body forces are needed to create the neutron SO gaps. While these gaps are not obtained when using microscopic effective interaction (see, for instance, the recent discussion by Smirnova et al [70]), the phenomenologically adjusted values of the two-body matrix element (TBME) or the explicit implementation of the three-body forces in SM approaches generate the expected neutron SO gaps.

The study of the nuclear structure around N = 28 has revealed, using the SM approach, that the tensor and two-body SO interactions are both at play to reduce the proton and neutron gaps far from stability. As proton orbits are degenerate at N = 28, these two components could not be disentangled. To separate these components, we propose in this section to study the evolution of p SO splitting between ³⁶₁₆S and ³⁴₁₄Si (a bubble nucleus). We find a sizable reduction of the SO splitting there, which is ascribed to the SO interaction (and not to tensor or central forces). This assumption comes from the fact that between these two nuclei protons are mainly removed from the 2s_1/2 (ℓ = 0) orbit, which does not exhibit a specific orientation between the orbital momentum and the intrinsic spin value.

In a second part, we propose to use the bubble nucleus ³⁴₁₄Si to test the density and the isospin-dependent parts of the SO interaction in MF approaches which have never been tested so far. These properties of the SO interaction are of minor importance in the valley of stability but are crucial at the neutron drip line and in superheavy nuclei.

As mentioned in section 3.1, the proton s_1/2 and d_3/2 orbits are degenerate at N = 28. The reduction of the p SO splitting at N = 28, between ⁴⁹Ca and ⁴⁷Ar, is due to the combined effects of proton–neutron interactions involving several monopole terms having potentially tensor and SO components. Applying equation (5) to the p orbits, the SO splitting is

$$\begin{eqnarray} \delta {\mathrm{SO}}({\rm p})&=& 1.33\left(V_{1{\mathrm{d}}_{3/2} 2{\mathrm{p}}_{1/2}}^{\mathrm{pn}} - V_{1{\mathrm{d}}_{3/2} 2{\mathrm{p}}_{3/2}}^{\mathrm{pn}}\right) \nonumber\\ &&+ 0.66\left(V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{1/2}}^{\mathrm{pn}} -V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{3/2}}^{\mathrm{pn}}\right). \end{eqnarray}\noindent \tag{ 10 }$$

The two contributions of this equation can be estimated from the study of the evolution of the p splitting in N = 21 nuclei. As the s_1/2 and d_3/2 proton orbits are separated by about 2.5 MeV (see ³⁵P in figure 5), one can therefore assume that these proton orbits are filled sequentially in the N ≃ 20 isotones. The pairing is providing a small dilution of occupancies among the two orbits as shown in [42] and figure 7. Between ⁴¹₂₀Ca₂₁ and ³⁷₁₆S₂₁, four protons are removed. The evolution of the [ν2p_3/2 − ν2p_1/2] SO splitting reads

$$\begin{equation} \delta {\rm SO}({\rm p}) \simeq 4 \left(V_{1{\mathrm{d}}_{3/2} 2{\mathrm{p}}_{1/2}}^{\mathrm{pn}} - V_{1{\mathrm{d}}_{3/2} 2{\mathrm{p}}_{3/2}}^{\mathrm{pn}}\right). \end{equation}\noindent \tag{ 11 }$$

By first looking at the evolution of the neutron [2p_3/2 − 2p_1/2] SO splitting between ⁴¹Ca₂₁ and ³⁷S₂₁ in figure 11, one investigates the role of proton–neutron interactions involving the 1d_3/2 protons. It is seen that the p SO splitting amounts to 2 MeV in the two nuclei. It follows that the SO splitting does not change between ⁴¹Ca and ³⁷S while four protons are removed from the d_3/2 shell, leading to δSO(p) ≃ 0 and V^pn_1d3/22p1/2 ≃ V^pn_1d3/22p3/2. We note that taking the full observed p strength, the conclusion does not change. Indeed, the SO splitting amounts to about 1.7 MeV in the two nuclei [59, 71]. We conclude that the spin-dependent part of the 1d–2p proton–neutron interaction is small. This is possibly due to the fact that the monopole terms V^pn_1d3/22p1/2 and V^pn_1d3/22p3/2 are weak in absolute value, as both their numbers of nodes and orbital momenta differ by one unit.

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From these observations, it was inferred also in [56] that the change in p SO splitting (δSO(p)) between ⁴⁹Ca and ⁴⁷Ar was ascribed to a modest depletion of the s_1/2 orbit by about 0.66 proton (see also section 5):

$$\begin{equation} \delta {\mathrm{SO}}({\mathrm{p}}) \simeq 0.66 \left(V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{1/2}}^{\mathrm{pn}} - V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{3/2}}^{\mathrm{pn}}\right). \end{equation} \tag{ 12 }$$

Thus at N ≃ 28 the effect of the two-body SO interaction is weak. The N = 21 region is then more propitious to study this interaction as more protons are depleted in the 2s_1/2 orbit between ³⁷S₂₁ and ³⁵Si₂₁.

As for ³⁵Si₂₁, energies and spectroscopic factors of its first 7/2⁻, 3/2⁻, 1/2⁻ and 5/2⁻ states were determined recently using the (d,p) transfer reaction in inverse kinematics at GANIL. They are compared to those of the isotone ³⁷S₂₁ [72] in figure 11. The observed p SO splitting changes by about 900 keV between ³⁷S and ³⁵Si while protons are removed from the 2s_1/2 orbit. This variation is observed in the major fragment of the single-particle strength, which contains correlations beyond the monopole terms. In order to unfold these correlations and to derive the change in monopole, the SM approach is used. While adjusting the monopole terms used in the SM calculations in order to reproduce the correlated experimental data [73], a decrease of the p SO splitting from 2 to 1.73 MeV is found for ³⁷S and an increase from 1.1 to 1.35 MeV is found for ³⁵Si. It follows that the change in [p_3/2–p_1/2] SO splitting (δSO(p)) amounts to about 25%, namely (1.73 − 1.35) = 0.38 MeV compared to the mean value (1.73 + 1.35)/2 = 1.54 MeV. This change is mainly ascribed to a change in the proton 2s_1/2 occupancy which is calculated to be Δ(2s_1/2) = 1.47:

$$\begin{equation} \delta {\mathrm{SO}}({\mathrm{p}}) \simeq 1.47 \left(V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{1/2}}^{\mathrm{pn}} - V_{2{\mathrm{s}}_{1/2} 2{\mathrm{p}}_{3/2}}^{\mathrm{pn}}\right) =380\,{\mathrm{keV}}. \end{equation} \tag{ 13 }$$

Note that by using the monopole terms (V^pn_2s1/22p1/2 and V^pn_2s1/22p3/2) derived in section 5 and figure 9, one finds using equation (13) a consistent change in the p SO splitting of 1.47(85 + 170) =  370 keV between ³⁷S and ³⁵Si.⁸ The present change in SO splitting is exclusively due to the two-body SO force, as monopole terms involving an ℓ = 0 component, as it does here with the s_1/2 orbit, do not contain any tensor component [30]. The only worry for this extraction of two-body term comes from the fact that the energy and wave function of 1/2⁻ state in ³⁵S, which is bound by about 0.5 MeV, may be influenced by the proximity of the continuum.

Having an orbital momentum ℓ = 0, the 2s_1/2 orbit is located in the center of the nucleus. With a weak 2s_1/2 occupancy, ³⁴Si is expected to have a depleted proton central density [74] in the SM as well as in the MF approaches. This depletion is also seen in the density profile calculated using the RMF/DDME2 interaction [75] in the left part of figure 12.

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On the other hand, the proton densities of the other N = 20 isotones do not display any central depletion (see the middle and the right part of figure 12). With this peculiarity of density depletion and isospin difference in central density (proton and neutron density profiles differ significantly), the ³⁴Si nucleus is a good candidate to study the SO interaction in MF approaches, as described below.

It is interesting to note that all theoretical approaches do not agree on the reason for the N = 28 shell disappearance. Li et al [18] say that the 'RMF model automatically reproduces the N = 28 spherical shell gap because it naturally includes the spin–orbit interaction and the correct isospin dependence of this term'. Therefore in the RMF approach 'there is no need for a tensor interaction to reproduce the quenching of the spherical N = 28 gap in the neutron-rich nuclei'. The bubble nucleus ³⁴Si will be used to ascertain these statements.

Relativistic mean-field (RMF) models introduced a description of the nuclear SO interaction [76] in terms of mesonic degrees of freedom, treating the nucleons as Dirac particles. Using a non-relativistic reduction of the Dirac equation, the SO term writes

$$\begin{equation} V_\tau^{\ell s} (r) = - [W_1 \partial_r \rho_\tau (r) + W_2 \partial_r \rho_{\tau\neq \tau '} (r)] \overrightarrow{\ell} \cdot \overrightarrow{s} \end{equation} \tag{ 14 }$$

in which W₁ and W₂ depend on the σ,ω,ρ meson coupling constants, and τ represents a proton or a neutron. Note that the W₁ and W₂ parameters have also density dependence in the RMF calculations that we neglect here. Thus beyond the ℓ·s term, the SO interaction contains a density dependence through ∂_rρ(r) and an isospin dependence through the W₁/W₂ ratio. Both relativistic and non-relativistic MF approaches agree on a significant density dependence of the SO interaction. However, while there is moderate isospin dependence of the SO interaction in the RMF approach (W₁/W₂ ≃  1), non-relativistic Hartree–Fock approaches have a large isospin dependence (W₁/W₂ = 2) provided by the exchange term [77] of the nuclear interaction. Beyond the fundamental importance to describe the SO interaction, the density and isospin dependences of the SO interaction have important consequences to model nuclei close to the drip line [78], to predict the isotopic shifts in the Pb region [79] and to describe superheavy nuclei which display a central-density depletion [80].

Having a significant proton-density depletion and a proton–neutron density asymmetry (there is no neutron central depletion), the ³⁴Si nucleus is a good candidate to study both the density and isospin dependences of the SO interaction as proposed in [72]. Indeed, RMF models give a large change in SO splitting by about 80% between ³⁷S and ³⁵Si, while MF models find a more modest change of about 30% for Δ(2s_1/2) = 1.5 [72].

The currently observed decrease of the SO splitting between ³⁷S and ³⁵Si by about 25% indicates that there is a density dependence of the SO interaction; otherwise no change in SO splitting would have been found. This modest change is in better agreement with theoretical models having a large isospin dependence, such as most of the MF models. Although extremely preliminary, this discovery would suggest that the treatment of the isospin dependence of the SO interaction in the RMF approach is not appropriate. If true this would have important consequences to model nuclei which are sensitive to the isospin dependence of the SO interaction. We give here two examples.

In some superheavy nuclei, a large proton and neutron central-density depletion may be present. They originate from the Coulomb repulsion between protons and from the filling of high-j neutron orbits which are located at the nuclear surface only [80]. Assuming no isospin dependence of the SO interaction as in the RMF approach, the proton and neutron central-density depletions mutually reduce the proton SO splitting for (ℓ = 1, 2 and 3) orbits which are probing the interior of the nucleus [80]. In this case, the Z = 114 gap, formed between the proton f_5/2 and f_7/2, would be significantly reduced to the benefit of a large shell gap at Z = 120,N = 172. Conversely, a rather large isospin dependence of the SO interaction would not lead to strong shell gaps [81] but rather to sub-shells the rigidity of which is further eroded by correlations. The second example deals with nuclei close to the neutron drip line. As they are expected to have a smoothened neutron density at their surface, the SO of high-j orbits is expected to be weakened.

In the RMF approach the SO interaction is weaker by about up to 65% as compared to MF approaches when reaching the drip line [78]. This change between RMF and MF models seems to be due to the different treatment of the isospin dependence of the SO interaction [78]. This change in SO interaction could have important consequences for modeling the evolution of the N = 28 shell closure far from stability as well as other SO magic numbers along which the r-process nucleosynthesis occurs [82]. We note here that other competing effects come into play to modify shell structure when approaching the drip line, as mentioned in the following sections.

In this section, we look whether the SO magic gaps 14, 50 and 82 are progressively vanishing when moving far from stability in the same manner as N = 28 does toward ⁴²Si. We start with the analogy of the evolution of the N = 14 shell gaps between ²²O and ²⁰C and pursue with the evolution of the SO magic number in the ¹³²Sn nucleus at N = 82. We show that, surprisingly, while the same forces are involved in all regions, the N = 82 gap remains very large contrary to the N = 14 and 28 ones. The ⁷⁸Ni nucleus, which lies in between the ⁴²Si and ¹³²Sn nuclei, is briefly discussed as well.

The N = 28 gap vanishes progressively by the removal of six protons from the doubly magic ⁴⁸Ca to the deformed ⁴²Si nucleus. As far as tensor forces are concerned, their action cancels in a spin saturated valence space, i.e. when orbits with aligned and anti-aligned spin orientations with respect to a given angular momentum are filled. In the present valence space for instance, the filling of the proton f_7/2 and f_5/2 annihilates the tensor effects on the filled d_5/2 and d_3/2 orbits. As the ⁴²Si nucleus is a spin unsaturated system in protons and neutrons, tensor effects are maximized, leading to reductions of both the proton d_3/2–d_5/2 and neutron f_7/2–f_5/2 splittings. Similar spin unsaturated nuclei exist in nature, such as (i) ²⁰C (πp_↑ and νd_↑ filled but not πp_↓ and νd_↓), (ii) ⁷⁸Ni (πf_↑ and νg_↑ filled but not πf_↓ and νg_↓) and (iii) ¹³²Sn (πg_↑ and νh_↑ filled but not πg_↓ and νh_↓). These examples are illustrated in figure 13. We would expect that the behavior of all these four mirror valence nuclei is similar, but this is not the case.

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Starting with the first example, N = 14, it was found in [83] that the energy of the first 2⁺ state drops by a factor of 2 between ²²O (3.2 MeV [84]) and ²⁰C (1.588 MeV). Then the reduction of the N = 14 gap, formed between the neutron d_5/2 and s_1/2 orbits, could be ascribed to the two monopole interactions involved when removing two protons from the p_1/2 orbit between ²²O and ²⁰C, i.e.

$$\begin{equation} \delta G^{\mathrm{pn}}(14) \simeq 2\left(V_{1{\mathrm{p}}_{1/2} 2{\mathrm{s}}_{1/2}}^{\mathrm{pn}} - V_{1{\mathrm{p}}_{1/2} 1{\mathrm{d}}_{5/2}}^{\mathrm{pn}}\right). \end{equation}\noindent \tag{ 15 }$$

We can estimate the reduction of the N = 14 gap by looking at similar forces in the valley of stability. The ordering of the d_5/2 and s_1/2 is exchanged between ¹⁷O (ground state 5/2⁺, 1/2⁺ at 870 keV) and ¹⁵C (ground states 1/2⁺, 5/2⁺ at 740 keV). Such a swapping of about 1.6 MeV between the two neutron orbits means that |V^pn_1p1/21d5/2| ≫ |V^pn_1p1/22s1/2|. The large value of the |V^pn_1p1/21d5/2| monopole can be ascribed to the attractive part of the tensor term. Assuming that the two monopoles are of similar intensity between A = 16 and 22, the N = 14 gap is expected to be reduced by 1.6 MeV between ²²O and ²⁰C, starting from a value of about 4 MeV in ²²O (see figure 10). This large reduction of the N = 14 spherical gap, by the removal of only two protons, can easily account for the decrease in 2⁺ energy in ²⁰C.

For the N = 28 shell gap, the removal of six protons is required to change drastically the nuclear structure. The effect of residual forces per nucleon is less important than in the lighter nuclei. It is also noticeable that the size of the N = 28 gap (4.8 MeV) is about 1 MeV larger than that of N = 14. These two effects, weaker monopole terms and increasingly larger SO splitting, could explain why the vanishing of the shell gap occurs further from stability at N = 28.

On the other hand, ¹³²Sn has all the properties of a doubly magic nucleus with a high 2⁺ energy of 4.04 MeV [85], due to a large shell gap [3]. Large spectroscopic factors are found in the ¹³²Sn (d,p) ¹³³Sn reaction [69, 86], which is a further confirmation that ¹³²Sn is a closed-shell nucleus. Note that the N = 82 gap is not strictly formed between the neutron h_11/2 and f_7/2 orbits, as the h_11/2 orbit lies below the d_3/2 one for Z = 50 [87]. This arises from the fact that the h_11/2–h_9/2 SO splitting is larger than the other SO splittings g_9/2–g_7/2, f_7/2–f_5/2 and d_5/2–d_3/2 in the N = 50, 28 and 14 regions, respectively. This increasingly larger splitting with ℓ value is due to the $\overrightarrow {\ell } \cdot \overrightarrow {s}$ term of equation (14).

In the N = 50 case, the rigidity of the ⁷⁸Ni nucleus with respect to quadrupole excitations is not known so far. Based on the study of spectroscopic information between Z = 30 and 38, the evolution of the N = 50 shell gap has been studied in [64]. A rather modest reduction of about 0.55 MeV was proposed between Z = 38 and 28. Nevertheless, as this reduction is combined with that of the Z = 28 gap [65], it is not absolutely sure that ⁷⁸Ni would remain spherical in its ground state.

Although similar tensor and SO forces are at play for N = 14, 28, 50 and 82, the corresponding shell closures do not have the same behavior when going far from stability. Quadrupole correlations dominate in N = 14 and 28 over the spherical shell gaps, while the N = 82 gap remains remarkably rigid in ¹³²Sn. This feature probably comes from two effects: (i) the SO shell gaps become larger for higher ℓ orbits and (ii) the monopole terms involved in reducing the shell gap are weakened in heavy nuclei because of the larger sizes of the orbits. In the next section, we explore the forces which come into play even further from stability.

Advances in radioactive beam productions offer the possibility to produce and study nuclei at the limit of particle stability. There, new forces are involved and the interaction with continuum states should progressively play a decisive role. We start this section with a qualitative analysis of the nuclear forces involved at N = 28 below ⁴²Si. We propose that this gap is further reduced, and illustrate our purpose by looking at recent experimental studies which evidence the inversion between the f_7/2 and p_3/2 orbits. We generalize this mechanism to other shell gaps, N = 50 and 82, and propose astrophysical consequences for the r-process nucleosynthesis below ¹³²Sn.

It was shown in [34, 88] that the neutron drip line extends further N = 28 in the Si and Al isotopic chains, i.e. at least up to ⁴⁴Si₃₀ and ⁴³Al₃₀, respectively. If the N = 28 shell gap is large, a sudden drop in S_2n would be found there and the drip line would lie at N = 28 and not beyond. To give an example, the neutron drip-line is exactly located at a magic shell in the O chain, where ²⁴O is bound by about 3.6 MeV and ²⁵O is unbound by about 770 keV [89]. So in this case, ²⁴O is a doubly magic nucleus [90–92] and the spherical shell gap is large enough to hamper the onset of quadrupole correlations. In contrast, the fact that the drip line extends further from stability at N = 28 is an indication of the presence of deformed nuclei which gain in binding energy due to correlations. This also hints at a further reduction of the N = 28 shell gap. This hypothesis is hard to prove in a direct way as it would require us to determine the size of the spherical shell gap there, which is out of reach so far, and which would not be an observable when nuclei are deformed. Therefore we shall use qualitative arguments as well as an example taken from another region of the chart of nuclides where similar forces are present to predict the behavior of the N = 28 shell gap further from stability. With this in mind, we will propose some extrapolations to other regions of the chart of nuclides.

Below ⁴²Si the change of the N = 28 shell gap between the 1f_7/2 and 2p_3/2 orbits is driven by the difference between the V^pn_1d5/22p3/2 and V^pn_1d5/21f7/2 monopole terms. On top of this monopole-driven effect, correlations play an important role in these nuclei. For these two monopole terms, the proton and neutron spin orientations are aligned with the orbital momentum, and their difference in angular momentum is one unit of ℏ (then the main part of the proton–neutron interaction comes from the central term of the nuclear force). On the other hand, the numbers of nodes in their wave functions are different. The fact that the 1f_7/2 and 1d_5/2 wave functions have the same number of nodes leads to a larger radial overlap of the wave functions, and a larger monopole term V^pn_1d5/21f7/2 as compared to V^pn_1d5/22p3/2. Consequently, when the 1d_5/2 orbit is completely filled, the 1f_7/2 orbit has a larger binding energy than that of the 2p_3/2 one. Conversely, when removing the six protons from the 1d_5/2 orbit, the neutron N = 28 gap further shrinks. Note that the calculations of [32] predict a decrease by 1.6 MeV down to ³⁶O₂₈. Of course these nuclei lie beyond the drip line, but this gives the monopole trend for the evolution of the N = 28 gap below ⁴²Si (see figure 14).

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In addition to this monopole effect, the nuclear-potential well will become more diffuse when moving towards the drip line. It was proposed by Dobaczweski et al [93] that this would bring the low-ℓ orbits more bound as compared to the high-ℓ ones, which could be viewed schematically as a reduced ℓ² term in the Nilsson potential. Then, an inversion of the two negative-parity orbits (f and p) could also happen, as suggested by several experimental works described below.

While the ν1f_7/2 orbit lies below the ν2p_3/2 orbit in most of the nuclei in the chart of nuclides (giving rise to the N = 28 shell gap), their ordering is reversed as soon as the πd_5/2 shell starts to empty, i.e. for Z < 14. The feature is well documented in the N = 15 and 17 isotones, not far from the stability valley. Indeed, the first negative-parity state of ^29,31₁₄Si_15,17 has I^π = 7/2⁻ while the 3/2⁻ state is located above. On the other hand, the situation is reversed in ₁₂Mg and ₁₀Ne isotopes, as described now. In the ²⁵Ne₁₅ nucleus [94], the 2p_3/2 orbit lies about 1 MeV below the 1f_7/2 orbit. This was determined by using the ²⁴Ne(d,p)²⁵Ne transfer reaction in inverse kinematics at the GANIL facility. Similar results were recently obtained at GANIL in ²⁷Ne₁₇, its first excited state at 765 keV having I^π = 3/2⁻ [95], the 7/2⁻ state being about 1 MeV above. Note that the assignment of the 3/2⁻ state was first proposed in [96, 97]. In the neighboring isotone ²⁹Mg, the second excited state at 1095 keV was observed from the study of the β decay of ²⁹Na. It could not be explained in the framework of SM calculations using the sd space, and consequently it was assigned an I^π = 3/2⁻ intruder state [98]. To our knowledge, two other experimental works are suggesting this inversion between the 1f and 2p orbits. By using the Coulomb break-up technique at RIKEN, Nakamura et al [99] found that the ground-state wave function of ³¹Ne was dominated by an ℓ = 1 component. Minomo et al [100] propose a contribution of a p_3/2 neutron using anti-symmetrized molecular dynamics calculations. Moreover, Wimmer et al argued that excitations to the 2p_3/2 (rather than the 1f_7/2) are required to account for the configuration of the 0⁺_1,2 states in ³²Mg determined through two-neutron transfer reaction at CERN/ISOLDE [101]. In these examples, in which the proton 1d_5/2 orbit is not yet fully filled, the N = 28 gap does not exist and the ν2p_3/2 orbit even lies below the ν1f_7/2 orbits. Conversely, this gives a further indication that the proton 1d_5/2 has to be filled to bind the ν1f_7/2 orbit enough to create an N = 28 gap which is present in heavier nuclei. Note that, from a theoretical point of view, this inversion was also proposed by Utsuno et al in 1999 [102].

We can reasonably assume that the present mechanism for the reduction of the N = 28 gap is robust, as it comes from the radial part of the wave functions and from the effect of a more diffused potential well at the drip line. As far as the first effect is concerned, it is also observed from the properties of the bare proton–neutron forces [103]. We therefore also expect the reduction of the N = 50 gap below ⁷⁸Ni and the reduction of the N = 82 gap below ¹³²Sn, as protons are removed from the 1f_7/2 and 1g_9/2 orbits, respectively (see figure 14). While the N = 82 gap is still present in the ¹³²Sn nucleus, it will be progressively reduced further from stability as ten protons are removed from the 1g_9/2 orbit to reach ¹²²Zr. The onset of quadrupole correlations across the N = 82 shell gap would occur if the neutron gap is reduced enough. This would bring a smoothening in the S_2n trend, as observed in figure 2 for the N = 28 isotones around ⁴⁴S. This has potentially important consequences for the location of the (n,γ)–(γ,n) equilibrium in the rapid neutron-capture process (r-process). This equilibrium occurs when neutron captures compete with photo-disintegrations. At this point, further neutron captures cannot occur as nuclei are immediately destroyed. Therefore the explosive process is stalled for a moment at such a waiting-point until β-decay occurs. When the drop in S_2n value is abrupt (for a large shell gap), the waiting-point nuclei are found at the closed shell. Then these nuclei are the main genitors of stable r-elements when decaying back to stability. When the shell gap is reduced, the location of r-progenitors is more extended and shifted to lower masses in a given Z chain, changing the fit of the r-process abundance curve accordingly (see, for instance, [82]).

To summarize this section, we propose that from both the nuclear two-body forces point of view and the MF point of view, a further reduction of the N = 28 shell gap will be occurring below ⁴²Si. By analogy, a reduction of the N = 50 and 82 shell gaps is anticipated, with possible important consequences for the r-process nucleosynthesis in the latter case.

In the previous sections we assumed that the two-body proton–neutron forces (or the monopole terms) remain of similar intensity when reaching the drip line. The SM description does not allow a self-consistent change of the monopole terms as a function of binding energy. However, the assumption of fixed monopole terms in a wide valence space is at best crude and probably wrong. Indeed, at the neutron drip line, valence protons are very deeply bound and their wave functions are well confined. On the other hand, neutrons, which are close to be unbound, have more dilute wave functions, and interactions with continuum states occur ([104] and references therein). It follows that the radial part of the wave functions of the valence proton and neutron have a weaker overlap, leading to a reduced effective interaction. One could ask: by how much is this interaction reduced?

The ₉F isotopic chain is a target choice to study changes in proton–neutron forces when approaching the drip line. Indeed, the drip line occurs at N = 16 in the ₈O chain, i.e. at ²⁴O, while the ³¹F₂₂ nucleus is still bound. Thus the addition of a single proton allows us to bind up to six neutrons more. It follows that, although occurring between nuclei showing large asymmetry in neutron and proton binding energy, the corresponding proton–neutron forces should be large enough to bind ³¹F₂₂.⁹ Up to ²⁹F₂₀, the same proton–neutron d_5/2–d_3/2 is expected to intervene. This strong monopole force is well known to produce significant changes closer to stability in the shell evolution of the N = 16 and 20 gaps. How is this force modified far from stability? The recent determinations of the binding energies of the (unbound) ²⁸F₁₉ [105] and the bound ²⁹F₂₀ [106] offer the possibility to start answering these questions. However, error bars on binding energies are still a bit too large at present to draw firm conclusions.

In this section, we propose to study the spectroscopy of the ²⁶₉F₁₇ to examine the proton–neutron d_5/2–d_3/2 interaction when approaching the drip line. We first explain the motivation for studying this nucleus and then use the new spectroscopic information required to study the proton–neutron force.

The ²⁶₉F₁₇ nucleus is a suitable choice for studying the proton–neutron d_5/2–d_3/2 interaction for at least three arguments: (i) it is bound by only 0.80(12) MeV, (ii) as the first excited state in ²⁴O lies at 4.47 MeV [91], ²⁶F can be viewed as a closed ²⁴O core plus a deeply bound proton in the d_5/2 orbit (S_p(²⁵F) ≃  –15 MeV) and an unbound neutron in the d_3/2 orbit (S_n(²⁵O) ≃ + 0.77 MeV [89]), (iii) the πd_5/2 and νd_3/2 orbits are rather well separated from other orbits which limit correlations of pairing and quadrupole origin. Binding energies BE(²⁶F)_J of the full multiplet of J = 1 − 4⁺ states arising from the πd_5/2⊗νd_3/2 coupling in ²⁶₉F₁₇ are needed to determine the role of the coupling to the continuum in the mean πd_5/2νd_3/2 interaction energy (int), defined in equation (16), as well as on the residual interaction which lifts the degeneracy between the components of the multiplet int(J) defined in equation (17). The mean interaction energy (int) reads

$$\begin{equation} {\mathrm{int}}= \sum\frac{(2J+1)\times {\mathrm{int}}(J)}{(2J+1)}. \end{equation} \tag{ 16 }$$

In this equation, int(J) term expresses the difference between the experimental binding energy of a state J in ²⁶F (BE(²⁶F)_J) and that of the ²⁴O + 1p + 1n system, BE(²⁶F_free), in which the valence proton and neutron do not interact. It reads

$$\begin{equation} {\mathrm{int}}(J)= {\mathrm{BE}}(^{26}\mathrm{F})_J - {\mathrm{BE}}(^{26}\mathrm{F}_{\mathrm{free}}), \end{equation} \tag{ 17 }$$

where

$$\begin{equation} {\mathrm{BE}}(^{26}\mathrm{F}_{\mathrm{free}}) = {\mathrm{BE}}(^{24}\mathrm{O})_{0^+} + {\mathrm{BE}}(\pi {\mathrm{d}}_{5/2}) + {\mathrm{BE}}(\nu {\mathrm{d}}_{3/2}). \end{equation} \tag{ 18 }$$

Assuming that

$$\begin{equation} {\mathrm{BE}}(\pi {\mathrm{d}}_{5/2})= {\mathrm{BE}}(^{25}\mathrm{F})_{5/2^+}-{\mathrm{BE}}(^{24}\mathrm{O})_{0^+} \end{equation} \tag{ 19 }$$

and

$$\begin{equation} {\mathrm{BE}}(\nu {\mathrm{d}}_{3/2})= {\mathrm{BE}}(^{25}\mathrm{O})_{3/2^+}-{\mathrm{BE}}(^{24}\mathrm{O})_{0^+}, \end{equation} \tag{ 20 }$$

we obtain

$$\begin{equation} {\mathrm{BE}}(^{26}\mathrm{F}_{\mathrm{free}}) = {\mathrm{BE}}(^{25}\mathrm{F})_{5/2^+}+{\mathrm{BE}}(^{25}\mathrm{O})_{3/2^+}-{\mathrm{BE}}(^{24}\mathrm{O})_{0^+}. \end{equation} \tag{ 21 }$$

Following the particle–particle coupling rule, the values of int(J) should display a parabola as a function of J in which |int(1)| and |int(4)| are the largest, as protons and neutrons maximize their wave-function overlap for these two values of total angular momentum, leading to the largest attractive interactions. The values of int(J) obtained from SM calculations (see the bottom part of figure 15) indeed form a parabola, the |int(3)| value being the lowest.

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The atomic masses of ^26,25F and ²⁴O were determined by Jurado et al [4]. They were used to determine the binding energy (BE) values used in the previous equations. The BE(²⁵O)_3/2⁺ − BE(²⁴O)_0⁺ = + 0.77 MeV difference is derived from the work of Hoffman et al [89]. The spin of the ground state of ²⁶F was determined to be I^π = 1⁺ by Reed et al [107] from the fact that I^π = 0,2⁺ states were populated in its β decay. The first-excited state of ²⁶F has recently been identified at 657(7) keV excitation energy by Stanoiu et al [108] at GANIL using in-beam γ-ray detection combined with the double fragmentation method. Moreover, an unbound state lying 270 keV above the neutron emission threshold has been proposed by Franck et al following a nucleon-exchange reaction at NSCL [109]. This state is likely to be the I^π = 3⁺ state. Finally, a long-lived J = 4⁺ isomer, T_1/2 = 2.50(5) ms, has been discovered at 642 keV by Lepailleur et al at GANIL [110]. It decays by an M3 internal transition to the J = 1⁺ ground state (86%), and by β-decay to ²⁶Ne or β-delayed neutron emission to ²⁵Ne. Gathering the measured binding energies of the J = 1 − 4⁺ multiplet in ²⁶₉F in figure 15¹⁰, we find that the components of the multiplet are experimentally more compressed than the SM predictions, whatever the effective interaction, USDA or USDB [111] . This would be due to a weaker residual interaction πd_5/2νd_3/2 close to the drip line. It would be interesting to compare these findings to calculations taking into account the effect of continuum. Note that the present conclusions rely strongly on atomic mass determination of the ²⁶₉F ground state, as well as the resonance at ∼270 keV we have assigned to the I^π = 3⁺ state of the multiplet. These two results deserve to be confirmed.