Magic and Doubly-Magic Nuclei

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Magic and Doubly-Magic Nuclei B. BLANK Centre d’Etudes Nucléaires de Bordeaux-Gradignan, Le Haut-Vigneau, F-33175 Gradignan Cedex, France P.H. REGAN Dept. of Physics, University of Surrey, Guildford, GU2 7XH, UK

A short history of the shell model An atomic nucleus is a complex arrangement of constituent neutrons and protons. An energetically bound nucleus is defined as one where all of these nucleons are bound against direct nucleon emission, with the limits of this condition defined by the proton and neutron drip-lines. In 1935, von Weizsäcker and Bethe proposed a semi-empirical mass formula describing the atomic nucleus as a liquid drop. This well-known formula introduces terms to account for volume, surface, Coulomb, asymmetry, and pairing energies which can be used to describe global properties of nuclei, such as binding energy and the limits of stability due to particle emission or fission. However, from the beginning, when this liquid-drop model was initially proposed, it was evident from experimental observations that this formula did not account for specific details of nuclear structure. In order to describe this fine structure correctly, additional microscopic effects had to be included in the modelling. Indeed, it was found that like electrons in the atom, the nucleons within an atomic nucleus are arranged in a shell-like structure. This shell structure was most evident in the additional binding of nuclei close to or at the socalled ‘magic numbers’ corresponding to proton or neutron numbers equal to 2, 8, 20, 28, 50, 82, and 126. After several attempts in the 1930s and 1940s, M. Goeppert Mayer as well as Haxel, Jensen and Suess recognized that the inclusion of a strong spin-orbit coupling interaction in the nuclear mean field was necessary to reproduce the observed shell ordering and associated magic numbers. Using a spinorbit force which binds the j=l+1/2 orbital stronger than the j=l-1/2 orbital, almost all the then known nuclear groundstate spins and magnetic moments of spherical nuclei could be explained in the framework of the “independent single-

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particle model”. This model assumes that properties like spins and magnetic moments of odd-A nuclei are due ostensibly to the orbital of the final, unpaired nucleon in the system, either a proton or a neutron. Also, due to the effect of the Pauli exclusion principle, an even number of nucleons will maximize their overlap by occupying identical, but time reversed orbits, which effectively cancels their influence on the nuclear structure. These theoretical and experimental findings and the apparent success of the shell model were some of the initial cornerstones of nuclear-structure physics. However, this early work was based almost totally on experimental information from β-stable nuclei (see Figure 1). Indeed, in one of her first articles on the “Spin-Orbit Coupling Model”, M. Goeppert Mayer used information of only two radioactive nuclei (35S and 129I) to justify her model. However, even today more than fifty years later, one of the most intriguing questions in nuclear physics remains whether or not the shell model and the associated magic numbers remain applicable for exotic nuclei far from the valley of β stability.

Figure 1. Chart of nuclei as a function of the neutron and proton number. β-stable nuclei are represented by black squares. Neutron-deficient nuclei, neutron-rich nuclei, α-particle emitters and direct proton emitters are also shown. The dashed lines indicate the predicted proton and neutron drip lines. The horizontal and vertical lines show the “classical” spherical magic numbers.

Nuclear Physics News, Vol. 10, No. 4, 2000

feature article Prior to discussing the current experimental status of the shell model and of magic numbers, we would like to survey the different ways that the magic numbers manifest themselves.

Shell effects in nuclear-structure physics As mentioned above, the shell structure of the atomic nucleus was initially inferred by the increased relative stability of nuclei with proton and/or neutron numbers at magic number values. This increased stability with respect to their neighbours is present in the mass and related binding energy of these isotopes. However, it is perhaps most visible in a oneneutron or two-neutron separation-energy plot, that is, the energy which has to be “pumped” into a nucleus to remove one or two neutrons. Figure 2 shows such a plot for the lead region, identifying the very pronounced discontinuity effect at neutron number 126. In a much simpler way, shell effects are also apparent in the increased numbers of beta-stable isotopes (isotones) with magic proton (neutron) numbers (see Figure 1). In particular, relative to their neighboring elements, extended chains of stable isotopes exist for calcium (40Ca to 48Ca), nickel (58Ni to 64Ni) and tin (112Sn to 124Sn) due to the effect of the magic numbers at Z=20, 28 and 50, respec-

tively. Similar effects can be found for the closed neutron shells at N=20, 28, 50, and 82. On the extreme proton-rich side of the valley of stability, where the proton-drip line (corresponding to the lightest particle-bound isotope of a given element) has been reached for elements up to nickel (Z=28), shell effects are responsible for increased stability. For Z=8, 20, and 28, the proton drip line extends considerably further in neutron number than for neighboring elements (Figure 1). The existence of the doubly-magic system, 48Ni, which is the nucleus with the largest proton excess observed to date, is due to shell effects. In an equivalent way, a subshell closure is most likely at the origin of the existence of the very proton-rich N=8 isotone, 22Si. A further indication of spherical shell effects can be found in the measured electric quadrupole moments of nuclei. As can be seen from Figure 2, these quadrupole moments are close to zero at the shell closures, which indicates that these nuclei have a spherical charge density distribution. In the same way, magic or doubly-magic nuclei are relatively difficult to excite which results in them having first excited states, which are noticeably higher in energy than for neighbouring nuclei. In return, the B(E2) value, a measure for the collectivity of a nucleus, becomes much smaller at shell closure.

Figure 2. Left-hand side: Two-neutron separation energy and α-decay Q values in the lead region around neutron number N=126. A pronounced shell effect is visible in both quantities (solid line). Right-hand side: Electric quadrupole moments for stable nuclei or nuclides close to stability. The vertical full lines indicate the shell closures at Z=8, 20, 28, 50, 82. The dashed line shows the subshell closure at Z=40, whereas the dotted line indicates the region where the N=82 shell is filled for stable nuclei. It can be seen from this plot that the shell closures at Z=28 and Z=40 are weaker than those at Z=20, 50, and 82.

Vol. 10, No. 4, 2000, Nuclear Physics News

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feature article Other hints to the shell structure are the relative abundance of certain even-even nuclides. One example is the high natural abundance for 56Fe, which results partially from the double closed shell of its beta-decay grandparent, 56Ni. Additional indicators include, βdecay half-lives (e.g. 56Ni compared to 57Ni or 55Co), the low neutron-capture cross section for nuclei with N=50, 82, and 126, the Q-values Qα for α emission (see Figure 2), and the fact that the heaviest stable nuclei in nature have proton numbers of Z=82 and/or neutron numbers of N=126. In addition, the increases in elemental abundance for stable nuclei with A~80 and A~130 are probably associated with r-process nucleosynthesis waiting points due to the N=50 and N=82 magic numbers, respectively. Here, the very neutron-rich nuclei with these neutron numbers are predicted to be bottlenecks for the production of heavier nuclei due to fast β decays and small neu-

tron separation energies. These abundance peaks in themselves point to the persistence of the N=50 and 82 magic numbers far from the valley of stability, a suggestion which is given further credence by the recent spectroscopic studies of nuclei around 78Ni and 132Sn (see later). Some of the findings described above appear to indicate that, to some extent at least, much of the well-established spherical shell structure persists far from stability. The relative stability of 48Ni is due to the stabilization for the Z=28 and the N=20 shell which seems to imply that, at least on the proton-rich side, the same shell structure is found as at the valley of stability. However, there are a number of recent reports of a breakdown of this ordering in neutron-rich systems. In addition, new deformed shell gaps are found at e.g. N or Z=40. In the following sections, we will discuss the evolution of nuclear structure with the isospin projection Tz = (N-Z)/2.

Figure 3. Yields of the N=20 isotones measured in an experiment to search for 28O [2]. The arrow shows the upper limit for the production of this isotope. The insert shows the results from the RIKEN experiment [3].

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feature article The Z=8 shell and the search for doubly-magic 28O of inversion’, which corresponds to a region of wellWith the exception of the α particle, 16O is the lightest deformed, neutron-rich nuclei around N=20. The coldoubly-magic nucleus. Its internal structure has been stud- lapse of the traditional shell structure was first observed ied for many years and is reasonably well understood. in 32Mg, where a comparatively low excitation energy of However, heavier oxygen isotopes might be expected to the first 2+ state compared to the lighter N=20 isotones give rise to other doubly- or semi-doubly-magic nuclei. The and a rather large B(E2) matrix element were measured term ‘semi-magic’ is sometimes used to describe a nucleus [4]. These experimental results indicated that this nuclewhere one set of nucleons, either protons or neutrons, fills us is no longer spherical and that the spherical N=20 a sub-shell rather than a full shell. Prominent sub-shell clo- shell becomes less dominant at large neutron excess. sures are known to exist at nucleon numbers 14, 40 and 64 A similar effect was also observed in the region around for example. Such a situation occurs for 22O, where, at 44S where the N=28 shell gap seems to vanish [5]. HowevN=14, the neutron d5/2 subshell is completely filled. How- er, another possible explanation could be the well-known ever, no experimental information is available for this shifts in the relative energies of the d3/2 and s1/2 proton nucleus. For its even-Z neighbour 24Ne, the excitation orbits between N=20 and N=28 [6]. energy of the first excited 2+ state rises by about 50%, The structure of magic nickel isotopes indicative of the subshell closure at N=14. The discovery of 48Ni represents the most recent step For the heaviest oxygen isotopes, it was long predicted that the effect of the N=20 shell might provide extra sta- in the investigation of magic nickel isotopes [6]. The bility to very neutron-rich oxygen isotopes. For this reason, known isotopes of this element span an exceptionally wide several experiments at GANIL, MSU, and RIKEN tried to range in mass, including the classical double shell closures synthesize and observe the doubly-magic 28O [1,2,3]. How- at 48Ni, 56Ni, and 78Ni as well as the N=40 subshell closure ever, it turned out that 24O was the heaviest stable nucleus for 68Ni. However, the detailed study of the structure of the with Z=8 protons (see Figure 3). This finding is particular- lightest and heaviest of these isotopes currently remains ly intriguing as for the heavier fluorine isotopes (Z=9), the elusive. Nonetheless, the very existence of 48Ni is most likeN=20 isotone 29F (and even 31F) is observed to be particle ly due to the stabilizing effect of the double shell gaps. In bound. One interpretation is that the double shell closure, the simple mass formula of von Groote et al. [7], these shell which would occur for 28O, prevents this nucleus from effects stabilize this nucleus by more than 1.3 MeV, which deforming away from a spherical shape. Therefore, it can not profit from any stabilizing effect of deformation, and consequently does not have enough binding energy to remain bound against direct neutron emission. In contrast, the spherical shape driving effect of the p shell closure at Z=8 has already vanished in 29F, allowing this nucleus to assume a more energetically favored, deformed shape. This does not mean that, in general, deformed nuclei are more tightly bound than spherical nuclei. Rather it expresses the fact that the additional degree of freedom of deformation helps nuclei to be more tightly bound. One might argue that the N=20 shell alone should prevent 29F from taking a deformed shape. However, a considerable amount of experimental evidence suggests Figure 4. Excitation energy of the first Iπ=2+ state in the nickel that the effect of the N=20 shell is reduced region. The shell effect at magic number N=28 is clearly visible for for neutron-rich systems, with deformed chromium, iron and nickel, whereas the subshell closure at N=40 ‘intruder’ configurations becoming favoured. [8,9] is only evident for nickel isotopes and disappears for iron as This effect gives rise to the so-called ‘island well as for zinc isotopes.


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feature article means that it would be highly unbound without shell effects. Therefore, it will be interesting to see how the N=20 shell influences observables such as the excitation energy of the first Iπ=2+ state in these very proton-rich systems. The energy of this first excited 2+ state and the small associated B(E2) value, which corresponds to the ease with which the state decays to a lower lying 0+ state, have been presented as indications for a subshell closure at N=40. N=40 is described as a “subshell” because the excitation energy necessary to excite a neutron to the next shell (in this case the g9/2 orbital), is only about half of the value compared to crossing the N=20 or N=50 shells. The excitation energies of the first 2+ states for iron, nickel, and zinc isotopes are shown in Figure 4. The shell effect at N=28 is clearly visible for all these isotopic chains, a convincing sign of a relatively strong shell closure for this neutron number. By contrast, the N=40 subshell is only readily visible for the nickel isotopes. For iron and zinc, the effect has already vanished, indicative of a relatively small shell effect. However, these results appear to be in contradiction with the recent observation of a small B(E2) value for 68Ni, which is interpreted as a sign of low collectivity and therefore high rigidity against collective excitation [10]. Nevertheless, the relatively small shell effect of the N=40 subshell is confirmed by spectroscopic investigations of neighboring nuclei where a strong core polarization effect is observed as soon as one proton or one neutron is added to the closed subshell nucleus 68Ni [11]. Figure 5 shows the excitation energy of the first 2+ state of the valence mirror nuclei of the nickel isotopes, the N=50 isotones and its neighbours [12]. The concept of the valence mirror assumes that the closed shells (Z=28 and N=50 in this case) form an inert core with the properties of the Z=28 isotopes and the N=50 isotones exclusively determined by the valence neutrons or protons, respectively. For example, 67Ni (49 neutrons and 28 protons) and 89Y (49 protons and 50 neutrons) should have the same structure in this simple picture. The experimental observation of a shell effect in the vicinity of the N=50 nucleus 90Zr is much more prevalent than around 68Ni. This is probably indicative of a stronger shell effect of the N=50 shell as compared to the Z=28 shell. Although the spectroscopy of the doubly magic system 78Ni remains tantalizingly out of current experimental reach, some information regarding the persistence of the magic proton and neutron cores in this highly neutron-rich nucleus has been recently obtained following a fragmentation study at GANIL, where both the energy of the yrast cascade and the lifetime of the deduced 8+ isomer are consistent with a doubly-magic core at 78Ni [13]. The E2 matrix element extracted from the lifetime of this isomer suggests that the 78Ni gap is present as expected,

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consistent with the increase in natural abundance of stable nuclei with A~80 due to r-process waiting points. As mentioned above 48Ni is probably a doubly-magic system. In the whole chart of nuclei, it is the only case of a doubly-magic nucleus with a bound mirror, 48Ca, doubly magic itself. The perspective to be able to study 48Ni in more detail may open the way for detailed studies of mirror symmetry in or near doubly-magic systems.

N=Z=40 and deformed shell gaps The apparent fragility of the N=40 subshell is surprising, since its Z=40 analog seems to persist over a wide range of neutron numbers. However, it is interesting to note that the spherical sub-shell effect appears to vanish for the self-conjugate, doubly semi-magic system 80Zr [14]. Classic in-beam spectroscopy measurements, originally performed at the Daresbury Laboratory, deduced a very large prolate deformation for this system of β≈0.4, based on the Grodzins estimate from the relatively small excitation energy of the first 2 + level. One explanation for the large distortion from spherical symmetry in this self-conjugate, ‘semi-magic’ system is the shape polarizing effect of a large deformed shell gap at nucleon number 38. It is thought that this drives the deformed configuration for 80Zr to a lower total energy than the semi-magic spherical solution. It has been proposed that the increased stability due to the deformed shell gaps which form the 76Sr core is also the reason for the particle stability of the very protonrich, odd-Z, Tz=-1/2 nucleus, 77Y. This is particularly surprising, since the lighter Tz=-1/2 analogs 69Br and 73Rb are found to be particle unbound [15]. If the simple, spherical mean field is allowed to deform into prolate and oblate shapes, new regions of low-level density occur at non-zero deformations, often corresponding to different major to minor axis ratios of the deformed mean-field shape. The observation of stable, highly deformed or superdeformed rotating nuclei has some foundation in these deformed shell gaps. For some nuclei, such as the N=Z=38 system, 76Sr, the deformed shell gaps result in a highly deformed ground-state configuration [16]. However, in other regions, at low values of angular momentum, the shell gaps are not so apparent and a different shaped minimum might be energetically favoured. These deformed ‘magic numbers’ become more apparent with increasing spin due to the varying effect of the Coriolis interaction on different orbitals with different spins and spin projections. This can be thought of as the analog of the spinorbit force which results in the spherical magic numbers. Cranked, deformed mean-field calculations show that


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Figure 5. Excitation energy of the first Iπ=2+ state in the N=50 region. The shell effect at the semi-magic number Z=40 persists also for N=48 and N=52. some large deformed shell gaps persist over wide ranges of rotational frequency. These deformed shell gaps can be related to various major to minor axis ratios for the deformed nuclear shape. Classic examples of ‘doubly magic’ superdeformed nuclei are 132Ce (Z=58,N=64), 152Dy (Z=66,N=86) which have major to minor axis ratios of approximately 3:2 and 2:1, respectively.

The Z=50, N=50 and 82 shell closures The magicity and stiffness of the Z=50 shell gap, representing the tin isotopes, is reflected in the remarkably constant value for the excitation energy of the first excited state in the even-even members of this isotopic chain. The excitation energy of the first 2+ states between 104Sn and 130Sn varies by less than 10% from the average value of approximately 1200 keV. However, even at this ‘stiffest’ of the shell gaps, competition from deformed two-particle two-hole or ‘intruder’ excitations can be observed at low excitation energies.

The location of the first 2+ state in the doubly-magic system 132Sn at a much increased excitation energy of around 4 MeV is consistent with the continuing effect of the N=82 shell gap, even at this high degree of ‘neutron richness’. Indeed, the localized peaks in the isotopic abundance of stable nuclei with A~80 and 130 has been interpreted as the effect of the N=50 and 82 shell gaps, respectively. The suggestion is that the extra binding due to these neutron numbers results in waiting points in the astrophysical r-process. Once these nuclei have β-decayed back to the valley of stability, they result in increased relative abundance. This suggests that the N=50 and N=82 neutron shell gaps remain significant even at large neutron excesses. The N=Z=50 nucleus, 100Sn, represents the heaviest self-conjugate doubly-magic system which remains particle bound. This nucleus has been identified to exist in fragmentation based experiments at both GSI and GANIL [17] and indeed its mass has been determined using the doublecyclotron method pioneered at GANIL [18]. Although the


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feature article spectroscopy of the excited states in 100Sn itself currently remains beyond experimental reach, there has been a significant body of work performed on neighboring nuclei, including the two-proton hole and two-neutron particle systems, 98Cd and 102Sn, respectively [19]. This work has enabled the effective proton and neutron charges for the core to be deduced which give a good understanding of the shell-model states around this core.

Heavier Shells and Super-Heavy Elements In recent years, there has been an explosion of data regarding the effect of the Z=82 shell gap. 208Pb is the heaviest, β-stable, doubly-magic system and as such has been studied extensively. This has enabled the observation of many of the single-particle energies and residual interactions required for sophisticated shell-model calculations to be performed. Higher spin states in nuclei around this region have shown that other collective effects, such as coupling to octupole phonon vibrations, should also be included for a full description of the states. State of the art work by the group at the University of Kentucky has probed the non-yrast states of this nucleus using (n,n’) style reactions which have resulted in candidate states for double-octupole phonon states [20]. However, as with other spherical closed shells, there is also evidence for competing deformed configurations, associated with particle-hole excitations across the spherical shell gap. Much progress has also been made recently in the spectroscopy of very neutron-deficient lead isotopes. Spectroscopic information is now available for nuclei ranging across 32 isotopes, from the N=100 nucleus 182Pb [21] up to the neutron-rich, N=130 system 212Pb [22]. The spectroscopy of the lightest isotopes of this element is experimentally hampered by competition from a large fission background, but useful information can be obtained using the technique of recoil-decay tagging on the fingerprint αdecay energies. The light lead isotopes show much interesting shape phenomena, with deformed minima competing with the near-spherical shape associated with the classic Z=82 shell closure. In all cases, the ground state configuration is dominated by the spherical shell gap. However, at low spins, a deformed prolate shape becomes energetically competitive, becoming yrast at the 2+ state. Indeed, for the mid-shell nucleus, 186Pb, (N=104 is exactly halfway between the N=82 and N=126 magic numbers), things are even more complicated with a third minimum associated with a deformed oblate shape being observed (see Figure 6). This makes an unusual case in nature of a quantal system whose ground state and first two excited states correspond

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to three separate minima associated with dramatically different macroscopic shapes [23-25]. These distinct minima seem to keep their individual characters up to significant spin, as determined by the observation of separate spherical and deformed or ‘K’-isomers in 188Pb [24]. On the neutron-rich side of 208Pb, experimental information is hampered by the lack of suitable reaction mechanisms for producing such nuclei. Some progress has been recently made using isomer spectroscopy following projectile fragmentation of uranium beams at the GSI laboratory, which have provided information on decays from isomeric states in this region [22]. Also, additional information can be obtained using partial fusion or deep-inelastic reactions to investigate slightly neutron-rich systems in this region, however, the exploration of sanctity of the Z=82 shell gap for very neutron-rich systems is currently out of experimental reach. Up to nucleon number 82, the spherical shell closures for protons and neutrons remain the same for both species of the nuclear fluid. There is, however, much current debate as to the location of the next proton magic number after Z=82. The strong Coulomb potential produces significant changes in the single particle potential compared to neutrons. The location of the next shell for both protons and neutrons is of some importance in predicting the long sought after ‘island of stability’ for superheavy elements. It has long been proposed that the stabilizing effect of a doubly-magic core would result in such a nucleus having a significantly increased decay half-life compared to its neighbours. This would be expected to arise due to an increase in the fission

Figure 6. Potential surface for 186Pb where three minima have been observed experimentally corresponding to three different deformations. Taken from reference [23], courtesy of Ramon Wyss.


feature article barrier, since one might expect there to be a resistance for a spherically doubly-magic system to deform. Depending on the parameterizations used, Skyrme-Hartree-Fock and relativistic mean-field type calculations [26] predict spherical shell gaps at proton numbers Z=114, 120 and 126, with neutron counterparts at N=172 and 184. There have been recent reports of α-decays from elements with Z=114, 116, and 118 from experiments at Berkeley and Dubna, however, at the time of going to press, neither result has been externally confirmed [27]. Compared to the doubly-magic superheavy nuclei predicted by theory, these nuclides are rather neutron-deficient. The relatively neutronrich nature of the candidates for the doubly-magic superheavy nuclei means that experimentally their study will most likely have to wait until the advent of intense, neutron-rich radioactive beams at near Coulomb-barrier energies.

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Conclusion and outlook The present overview highlights the effect which closed shells have on nuclear structure. These shell effects, first established for nuclei close to the valley of stability, persist in many regions also far from stability. This is particularly true for doubly-magic nuclei as demonstrated by the instability of 28O, the particle stability of 48Ni or the structure of nuclei in the vicinity of doubly-magic 78Ni. However, evidence for the persistence or disappearance of shell effects is often still scarce. The advent of new powerful machines to produce nuclei far from stability with much higher intensity accompanied by the continuous evolution of computing power for an improved modeling of the atomic nucleus will certainly lead to a deeper understanding of nuclear structure far from stability in the up-coming decade.

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