Shell Model Spectroscopy Far from Stability

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Shell model spectroscopy far from stability

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* Shell model spectroscopy far from stability

A Poves

Departamento de Física Teórica and IFT, UAM-CSIC, Universidad Autónoma de Madrid, Spain

E-mail: [email protected]

Received 24 November 2016, revised 1 June 2017 Accepted for publication 6 June 2017 Published 29 June 2017

Abstract I will discuss the impact of the research on exotic nuclei carried out at ISO- LDE and later on, following its impulse, in many laboratories worldwide, in our present knowledge of the nuclear dynamics. I will put particular emphasis on the occurrence of islands of inversion (IoI) at several magic numbers for the neutrons, far from stability. I will also discuss the appearance of new local shell closures in the calcium and silicon isotopic chains, and qualify their status as new doubly magic nuclei.

Keywords: nuclear structure, shell model calculations, neutron rich nuclei, new magic numbers, islands of inversion

(Some ﬁgures may appear in colour only in the online journal) 1. Introduction

Not so long ago, experimental nuclear spectroscopy was dominated by heavy ion collisions and high spin physics. Mean ﬁeld methods thrived in the theoretical arena, later joined by the interacting boson model. The shell model realm seemed dormant. Nevertheless, studies of very neutron rich radioactive isotopes were pursued at the ISOLDE facility under the CERN umbrella. And one can safely say that this activity paved the way for the present explosion of new results on nuclei far from stability, which have radically modiﬁed our understanding of nuclear dynamics. In this contribution, I will deal with the medium–light neutron rich nuclei from oxygen to calcium with special emphasis on the contributions from ISOLDE.

2. The discovery of the N=20 ‘island of inversion’

The story starts with the mass measurements made at CERN by what would later be the Orsay group of the ISOLDE Collaboration [1]. They found an excess of binding in 31Na and 32Na which could not be explained by the available theoretical models. Simultaneously, a group of

* This article belongs to the Focus on Exotic Beams at ISOLDE: A Laboratory Portrait special issue.

Original content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves theorists at Orsay tried to reproduce this effect with projected Hartree–Fock (PHF) calculations using density dependent interactions [2]. They got a normal, spherical, solution for 31Na with a small shoulder in the prolate side of the energy versus deformation curve. Never- theless, they argued that by applying a rotational correction, this shoulder might become a minimum, and explained that the anomaly was due to the gain in energy due to deformation. Even with this proviso, their model was unable to explain the 31Na data. Three decades and huge developments in beyond mean field (BMF) techniques were necessary to produce this minimum in the easier to treat nucleus 32Mg. All in all, their idea was correct even if the calculations were not conclusive. Already in Thibault’s paper [1], the possibility of having a new region of deformation was contemplated. Further work by the same group [3–5] extended the mass measurements to magnesium isotopes finding the same kind of anomaly in 32Mg. In addition, studying the beta decay of 32Na, they populated the first 2+ state in 32Mg at 881keV. Indeed this is a very low excitation energy compared with the corresponding one in 30Mg (1.48 MeV). Much more so, common sense would predict that the 2+ excitation energy should increase at the magic neutron number N=20 and not decrease almost by a factor two. The idea of the existence of an unexpected new region of deformation, precisely where the current models would have predicted enhanced sphericity, was reinforced. New shell model studies of these nuclei were triggered by a paper by Wildenthal and Chung [6] in 1980 with the provocative title ‘Collapse of the conventional shell-model ordering in the very-neutron-rich isotopes of Na and Mg’. In it, the authors, using their newly fitted high quality interaction for the upper part of the sd-shell, concluded that it was impossible to explain the behaviour of the neutron rich N=20 isotones in this valence space. The question that remains unanswered, and which has continued to produce a lot of confusion until now, is the meaning of the sentence ‘Collapse of the conventional shell-model ordering’. For many, this had to be interpreted as if, at the mean field level, some single particle orbits of the pf-shell would lay below the upper orbits of the sd-shell. Work in this direction was carried out by the Glasgow group [7, 8] by including the 0f7/2 orbit in the valence space, essentially degenerated with the 0d3/2. With this prescription they indeed got an increase in the binding energy of the N=20 isotones, but the proposed solution was far from being physically sound, and did not address at all the issue about deformation risen by the PHF calculations on [2]. Our work in [9] interpreted the collapse of the shell model ordering in a different way: they were not the spherical single particle orbits of the two adjacent major harmonic oscillator shells which collapsed far from stability, instead, what happened was that the intruder configurations of (neutron) 2p–2h nature, somehow became more bound than the normal, 0p–0h, ones. Therefore, the collapse must be due to the nuclear correlations. This interpretation makes a natural link with the surmise of [2], provided the correlations are of quadrupole– quadrupole type and the intruder configurations become deformed. The problem was how to get this behaviour out of a realistic shell model calculation. The solution was to include the 1p3/2 orbit—the quadrupole partner of the 0f7/2—in the valence space. This calculation produced deformed solutions for 32Mg, 31Na and 30Ne, keeping a substantial single particle gap between the sd and the pf-shells. Shortly after, Warburton, Becker and Brown [10] made extensive shell model calculations in the same model space reaching the same conclusions. And they coined the term ‘island of inversion’ (IoI) which has enjoyed immense success since then (I would have favoured the alternative ‘Island of Deformation’ though). Further work in the same context was carried out in [11, 12]. Since then, there has been a plethora of new experimental results, including mass measurements, radii isotope shifts, magnetic and quadrupole moments, spectroscopic factors

2 J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves etc, many of them performed at ISOLDE [13, 14]. The existence of an IoI at N=20 is now well established, even if its boundaries are not fully delineated yet. A very important mile- stone was the measurement during a coulex experiment at RIKEN of the B(E2) connecting the 2+ with the ground state of 32Mg, which conﬁrmed its well deformed nature corresponding to β=0.45 [15]. Other coulex experiments followed soon [16]. Indeed, the activity in the theory sector continued, both in the shell model framework [17] and with BMF methods [18]. The latter only got quantitative agreement with experiment after including the triaxial degrees of freedom, breaking time reversal invariance and allowing conﬁguration mixing via the generator coordinate approximation [19]. A more detailed description of the mechanism producing the IoI can be found in [20].

3. The heaviest potassium and calcium isotopes

We have seen how the availability of very heavy sodium isotopes at ISOLDE made it possible to discover the N=20 IoI. But many other elements were abundantly produced as well, and put to good use. This is the case for potassium, studied mainly by the Strasbourg members of the ISOLDE collaboration. 52K was produced for the first time (hence discovered) at ISO- LDE. Its decay produced and discovered 52Ca as well [21], whose spectroscopy was very interesting, with a first 2+ excited state at 2.56MeV, a rather high energy. However, the authors did not dare to claim that it was, as nowadays is accepted, a new doubly magic nuclei, which anticipated the appearance of new magic numbers far from stability. In retrospect, they lacked the propaganda skills. Only recently has 52Ca come back on the scene, fostered by the debate about the doubly magic character of 54Ca. Two new ISOLDE experiments related to it have been published in Nature, one exploring the masses [22] and another the radii isotope shifts [23] of the heaviest calcium isotopes. Previously, the radii shifts of the potassium isotopes beyond N=28, measured at ISOLDE as well, had been published in [24]. The work carried out in [23] extends our knowledge of the calcium radii up to 52Ca. Its basic result is that beyond N=28 and up to N=32, the radii increase linearly, with a slope which is far larger that all the available theoretical predictions (by the way, the same behaviour holds in the potassium isotopes [24]). That the proton radius increases with the neutron excess is well understood, and is due to the mechanism of equalization of neutron and proton radii; the problem is why the slope changes so much when neutrons start filling the orbit 1p3/2. A very appealing explanation of this behaviour has been presented in [25], which surmises that the radius of the p orbits is much larger than expected, rather halo-like, independently of their proximity to the neutron emission threshold. Obviously this mechanism is not directly related to the doubly magic nature of 52Ca, because it only depends of the occupancies of the p orbits. This has clearly been overlooked by the authors of [23], who write (sic): ‘The large and unexpected increase of the size of the neutron-rich calcium isotopes beyond N=28 challenges the doubly magic nature of 52Ca and opens new intriguing questions on the evolution of nuclear sizes away from stability, which are of importance for our understanding of neutron-rich atomic nuclei’. In fact, the linear increase of the radius of 52Ca is due to its doubly magic nature. At odds with the statement above, only a very large occupancy of the 0f5/2 orbit, i.e. a non magic configuration, can produce a smaller radius of 52Ca. Therefore, its (local) doubly magicity is not at stake. It is more so in view of the measure of the mass of 54Ca in [22] which shows a clear change of slope in the S2n curve between N=32 and N=34. Now, we turn to 54Ca itself. How do we interpret the experimental value [26] of the excitation energy of its first 2+, at 2.04MeV? Among the several candidates to doubly magic

3 J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves

+ Figure 1. Experimental excitation energies of the first 2 states in the calcium isotopes. For N=12 and N=14 we take the energies in the mirror nuclei 32Mg and 34Si and for N=36 the prediction of the KB3GR interaction (crossed circles). in the calcium isotopes, 34Ca (3.3 MeV, taken from its mirror 34Si), 36Ca (3.1 MeV), 40Ca (3.9 MeV), 48Ca (3.8 MeV) and 52Ca (2.6 MeV), it has the lowest 2+.In56Ca, the 2+ is predicted by the KB3GR interaction at 0.92MeV. Therefore, at N=34 there will be no peak in the 2+ energy signalling a magic neutron number, just a marked shoulder, similar to the one at N=16, corresponding to the filling of the orbit 1s1/2. This is better seen with in figure 1. Nonetheless, all the shell model calculations with different effective interactions predict that the ground state of 54Ca is dominated to better than 90% by the neutron con- 8 4 2 figuration (0f7/2) :(1p3/2) :(1p1/2) (and that independently of their prediction for the excitation energy of the 2+). Should we give it the doubly magic label, or decide that it only shows a local closure of the 1p1/2 orbit favoured by the weak neutron neutron interaction between these orbits? Concerning the potassium isotopes, the studies at ISOLDE were fundamental to understand the evolution of the effective single particle energies (ESPE) of the proton orbits 1s1/2 and 0d3/2 when the neutron number increases from N=20 to N=28 and above it. As is seen in figure 2 at N=20, the 0d3/2 orbit is almost 3MeV above the 1s1/2. As neutrons are 47 added in the orbit 0f7/2 their splitting at the ESPE level decreases linearly, and in Kat N=28 they are inverted and quasi degenerated. When neutrons start filling the 1p3/2 orbit the effect is the opposite, and the normal ordering is re-established [28]. This result predates the discussions of the ‘shell evolution’ [30] in terms of the different components of the monopole hamiltonian (central, tensor and spin-orbit) and, in particular, plays a crucial role in the structure of the very neutron rich N=28 isotopes. Notice that the same behaviour shows up at N=28 in 45Cl and 43P confirming the monopole nature of the effect [29].

4. Shape coexistence in 32Mg and 34Si

Another highlight of ISOLDE in the pioneer days was the discovery of the doubly magic nature of 34Si [31]. This time the claim was prominent in the title. However, neither the fact that the proton number Z=14 behaves as a new magic number in combination with N=20,

4 J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves

+ + Figure 2. Evolution of the splitting between the lowest 1/2 and 3/2 states in the potassium, chlorine and phosphor isotopes. Comparison of the experimental data with the predictions of the SDPF-U interaction [27].

Figure 3. Level schemes of 32Mg and 34Si, experimental data compared with the SM-CI results using the SDPF-U-MIX interaction. nor the very abrupt change from double magicity in Z=14 and N=20 to deformation in Z=12 and N=20 stirred a lot of attention at the moment. Nevertheless, two questions arose: where is the spherical 0+ state in 32Mg? And where is the deformed 0+ state in 34Si? Many experimental searches gave null or inconclusive results, and we had to wait almost twenty years to have the answers. + The first excited 0 state in 32Mg was finally found at ISOLDE by K Wimmer et al in 2010 at 1.06MeV [32]. Soon after, the long sought excited 0+ state in 34Si was discovered at GANIL by Rotaru et al at 2.7MeV [33]. The experimental situation is now depicted in figure 3, compared with the shell model with configuration interaction, SM-CI, results using the SDPF-U-MIX interaction [20]. The intriguing point is that according to the calculations,

5 J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves whereas the excited 0+ state in 34Si is indeed the head of a rotational band of 2p-2h nature and the shape coexistence picture holds nicely, the situation in 32Mg is more complicated, due to the mixing of super-deformed intruder states of 4p-4h nature with the deformed and spherical ones. It turns out that the ground state is 9% 0p-0h, 54% 2p-2h, 35% 4p-4h and 1% 6p-6h, thus, it is a mixture of deformed and superdeformed prolate shapes and rather than of shape coexistence it makes sense to speak of shape mixing. However, the excited 0+ has 33% 0p-0h, 12% 2p-2h, 54% 4p-4h and 1% 6p-6h, is a surprising hybrid of spherical and superdeformed shapes, whose direct mixing matrix element is strictly zero. Clearly, it is not a case of shape mixing, could it be an example of shape entanglement? The cross sections to the two 0+ states in 32Mg measured in [32] were hardly compatible with an schematic two state shape coexistence model [34]. However, a three state description, inspired in the full SM-CI results has been recently shown to be able to explain the experimental data [35].

5. N=28, from 48Ca down to 42Si and 40Mg

The interest in the very neutron rich N=20 isotones was soon extended to the N=28 region. There were indications of a weakening of the N=28 closure in 44S which has a very low 0+ isomer above the first 2+ state [36], but, almost simultaneously, it was claimed that 42Si was doubly magic [37], although a disclaimer followed soon [38]. The dispute was settled with the publication by Bastin et al of their GANIL results [39], which located the 2+ energy at 0.77MeV. New experimental data confirmed the evidence in favour of a new IoI at N=28 [40, 41]. Shell model calculations predict oblate deformation for 42Si and prolate deformation for 40Mg [27, 42]. A key ingredient to get deformed intruder states at N=28 is the degeneracy of the 1s1/2 and 0d3/2 proton orbits, which enhances the quadrupole col- lectivity via the pseudo-SU(3) coupling scheme. The shell model calculations of [20] go a step further, proposing that the IoI’satN=20 and N=28 merge, at least in the magnesium isotopes. This is supported by the new experimental results at RIKEN [43], including the 2+ and 4+ energies up to 38Mg, and the corresponding B(E2) transition probabilities. It is seen in figure 4 that the excitation energies of the 2+ states remain very low from N=20 to N=26 and agree perfectly with the calculations. Therefore, the extrapolation to N=28, and N=30 is guaranteed. Indeed, there is no sign at all of magicity in the magnesium chain, neither at N=20 nor at N=28. The measured (and calculated) B(E2)ʼs are very large and correspond to b ~ 0.4. The 4+ energies follow nicely the J(J+1) law. Thus, we can safely speak of a chain of well deformed nuclei extending from 32Mg to 40Mg or even to 42Mg, depending on which turns out to be the last bound magnesium isotope. In the silicon chain we observe the persistence of the N=20 closure in doubly magic 34Si and the vanishing of the N=28 closure in 42Si, which belongs to the N=28 IoI. Only in the calcium isotopes, both the N=20 and the N=28 closures survive.

6. Conclusion

I have tried to show how the experimental studies of nuclei far from stability, pioneered by ISOLDE, have shaped our modern view of nuclear dynamics. These experiments have brought to light many important concepts, among others; the shell evolution governed by the monopole Hamiltonian; the overwhelming importance of nuclear correlations; the ubiquity of the phenomena of shape coexistence and shape mixing and the proliferation of IoI driven by the quadrupole correlations.

6 J. Phys. G: Nucl. Part. Phys. 44 (2017) 084002 A Poves

+ Figure 4. Excitation energies of the 2 states of the calcium, silicon and magnesium isotopes from N=18 to N=30. Experimental results and calculated values with the SDPF-U-MIX interaction.

Acknowledgments

The shell model results discussed in this article were obtained in collaboration with E Caurier and F Nowacki. My interest in the physics of exotic nuclei originates in the collaboration with Guy Walter. My thanks go to all three. This work is partly supported by MINECO (Spain) grant FPA2014-57196 and Programme ‘Centros de Excelencia Severo Ochoa’ SEV-2012-0249.

ORCID

A Poves https://orcid.org/0000-0001-7539-388X

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