595_e magic numbers at N=20 and 28 fade away with neutron number and the new magic numbers at N=14, 16, and 32 seem to appear

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595_e magic numbers at N=20 and 28 fade away with neutron number and the new magic numbers at N=14, 16, and 32 seem to appear



     Shell Structure of Exotic Nuclei

     J. Dobaczewski,1 N. Michel,2,3 W. Nazarewicz,1?3 M. Ploszajczak,4 J. Rotureau2,3 Institute of Theoretical Physics, Warsaw University ul. Ho?B a 69, 00-681 Warsaw, Poland z 2 Department of Physics & Astronomy, University of Tennessee Knoxville, Tennessee 37996, USA 3 Physics Division, Oak Ridge National Laboratory P.O. Box 2008, Oak Ridge, Tennessee 37831, USA 4 Grand Acc?äl?ärateur National d??Ions Lourds (GANIL) ee CEA/DSM - CNRS/IN2P3, BP 55027, F-14076 Caen Cedex, France February 8, 2008

     Abstract Theoretical predictions and experimental discoveries for neutron-rich, short-lived nuclei far from stability indicate that the familiar concept of nucleonic shell structure should be considered as less robust than previously thought. The notion of single-particle motion in exotic nuclei is reviewed with a particular focus on three aspects: (i) variations of nuclear mean ?eld with neutron excess due to tensor interactions; (ii) importance of many-body correlations; and (iii) in?uence of open channels on properties of weakly bound and unbound nuclear states.


     arXiv:nucl-th/0701047v1 17 Jan 2007



     Shell structure is a fundamental property of ?nite Fermi systems and atomic nuclei are not exceptions. In fact, the very concept of single-particle motion is a cornerstone of nuclear structure [1]. The stability of nuclei is rooted in non-uniformities of the single-particle level distribution and presence of magic gaps, and those can be traced back to classical periodic orbits of a nucleon moving in a one-body potential [2]. The shell e?ects are, therefore, intimately related to the mean-?eld approximation, to which the very notion of individual particle orbits is inherent. The average potential of the nuclear Shell Model (SM), characterized by a ?at bottom, a relatively narrow surface region, and a strong spin-orbit term, was introduced in 1949 by Maria G??ppert-Mayer o [3], Otto Haxel, Hans Jensen, and Hans Suess [4]. This SM concept, illustrated in the left-hand-side diagram of Fig. 1, explains the nuclear behavior in terms of single nucleons moving in individual orbits. But how robust is this picture? In nuclei close to the beta stability line, the modern nuclear SM, in which the dominant one-body behavior is augmented by a two-body residual interaction, is a very powerful tool [7]. However, a signi?cant new

    theme concerns shell structure near the particle drip lines and in the superheavy nuclei. Theoretical predictions and experimental discoveries in the last decade indicate that nucleonic shell structure is being recognized now as a more local concept [8, 9, 10]. The experimental data indicate that the magic numbers in neutron-rich nuclei are not the immutable benchmarks they 1

     1949 Nuclear Shell Structure


     h9/2 f7/2

     p1/2 f5/2 i13/2

     d3/2 h11/2 s1/2 g7/2 d5/2


     Figure 1: The cornerstone of nuclear structure for over half a century has been the shell model of G??ppert-Mayer and Jensen, in which each nucleon is assumed to move in average potential. The o left-hand-side diagram shows the shell structure characteristic of nuclei close to the valley of stability. The combined e?ects of ?at bottom and small di?useness of the mean-?eld potential, together with a strong spin-orbit term, result in a sizeable angular-momentum splitting of single-particle levels that yields the experimentally observed shell and subshell closures. The right-hand-side diagram shows schematically the shell structure that may exist in neutron-rich nuclei, which corresponds to a more uniform distribution of energy levels and the quenching of known magic gaps. (Based on Refs. [5, 6].) were once thought to be [11]: the magic numbers at N=20 and 28 fade away with neutron number and the new magic numbers at N=14, 16, and 32 seem to appear. Why is shell structure changing in the neutron-rich environment? There are several good reasons for this. First, as discussed in Sec. 2, the nuclear mean ?eld is expected to strongly depend on the orbits being ?lled due to the tensor components of e?ective interaction. Second, many-body correlations, such as pairing, involving weakly bound and unbound nucleons become crucial when the neutron separation energy Sn is small. Indeed, as seen from the approximate relation between the neutron Fermi level, ?Ën , neutron pairing gap, ?n , and Sn [12, 6]: Sn ?Ö ??Ën ? ?n , (1)

     in the limit of weak binding, the single-particle ?eld characterized by ?Ën and many-body correlations represented by the pairing ?eld ?n become equally important. In other words, contrary to the situation encountered close to the line of beta stability, con?guration-mixing e?ects in loosely bound nuclei near the drip lines can no longer be treated as a small perturbation atop the dominant mean ?eld; they 2

     & %$? ?ì ?ê?è " "?ì|' ?|?ê

     ?ê???ì|?? ?è ?ê?é??


     h9/2 f5/2 p1/2 p3/2 f7/2



     h11/2 g7/2 d3/2 s1/2 d5/2



     (? ' ? & %$? ?ì?è?ì?é# |?ê?è " !? ?ê?è?è ?ê ?|

     are of primary importance for the very existence of these systems. This means that in the limit of small Sn the notion of a single-particle motion in a mean ?eld, the basic idea behind the nuclear SM, is no longer a viable ansatz. Third, the nucleus is an open quantum system (OQS). The presence of states that are unbound to particle-emission may have signi?cant impact on spectroscopic properties of nuclei, especially those close to the particle drip lines. In its standard realization [7, 13], the nuclear shell model assumes that the many-nucleon system is perfectly isolated from an external environment of scattering states and decay channels. The validity of such a closed quantum system (CQS) framework is sometimes justi?ed by relatively high one-particle (neutron or proton) separation energies in nuclei close to the valley of beta stability. However, weakly bound or unbound nuclear states cannot be treated in a CQS formalism. As discussed in Sec. 3, a consistent description of the interplay between scattering states, resonances, and bound states in the many-body wave function requires an OQS formulation. Figure 2 schematically illustrates di?erent physics components that are important for the structure of neutron-rich nuclei.

     Open QS

     Sn=0 Energy

     Correl on ati dom i nated:



     te ta



     Closed QS

     n ou Gr

     r- process



     N eutron num ber

     Figure 2: Schematic diagram illustrating various aspects of physics important in neutron-rich nuclei. One-neutron separation energies Sn , relative to the one-neutron drip-line limit (Sn =0), are shown for some isotopic chain as a function of N. The unbound nuclei beyond the one-neutron drip line are resonances. Their widths (represented by a

    dark area) vary depending on excitation energy and angular momentum. At low excitation energies, well-bound nuclei can be considered as closed quantum systems (QS). Weakly bound and unbound nuclei are open quantum systems that are strongly coupled to the environment of scattering and decay channels. The region of particularly strong many-body correlations, Sn <2 MeV or |?Ën | ?Ö ?n (1), is indicated. The astrophysical r-process is expected to proceed in the region of low separation energies, around 2-4 MeV [14]. In this region, e?ective interactions are strongly a?ected by isospin, and the many-body correlations and continuum e?ects are essential. There are many examples of the impact of many-body correlations and continuum coupling on structural properties of neutron-rich nuclei. Halos with their low-energy decay thresholds and cluster structures are obvious cases [15]. The islands of inversions around magic numbers N=20 and 28 [16, 17, 18] o?er another example. In those nuclei, dramatic asymmetry between proton and neutron Fermi 3

     surfaces gives rise to new couplings and the explanation involves coexistence e?ects due to intruder states and modi?cations of the e?ective interaction. In the following sections, various aspects of physics responsible for structural (and spectroscopic) changes in exotic nuclei are brie?y reviewed. Two detailed examples are given. Section 2 is devoted to e?ective interactions and shell structure in neutron-rich nuclei. The impact of the continuum coupling on unbound states in neutron-rich systems is discussed in Sec. 3.


     E?ective Interactions in Exotic Nuclei

     The new experimental advances along the isospin axis and towards the territory of super-heavy elements at the limit of mass and charge require safe and reliable theoretical predictions of nuclear properties throughout the whole nuclear chart. The tool of choice is the nuclear density functional theory (DFT) based on the self-consistent Hartree-Fock-Bogoliubov (HFB) method. The key component is the universal energy density functional (EDF), which will be able to describe properties of ?nite nuclei as well as extended asymmetric nucleonic matter. The development of such a functional, including dynamical e?ects and symmetry restoration, is one of the main goals of the ?eld [19, 20]. Developing a nuclear EDF requires a better understanding of the density and gradient dependence, spin e?ects, and pairing, as well as an improved treatment of symmetry-breaking e?ects and many-body correlations. Since the nuclear DFT deals with two kinds of nucleons, the isospin degree of freedom has to be introduced, and the isoscalar and isovector densities have to be considered [21]. A topic, relevant to the spin-isospin channel, that has recently become the focus of considerable attention is the in?uence of the tensor

    interaction on single-particle levels. It is discussed below.


     Spin-Orbit Splitting and Tensor Interaction

     Positions of single-particle levels change with varying neutron or proton numbers. These changes may lead to dramatic variations in properties of nuclei. For example, a shift in position of the neutron 1f5/2 level has been recently proposed to explain the appearance of a new N=32 magic gap in neutron-rich isotopes above Z=20. This is only one of several such examples recently identi?ed in light nuclei and interpreted within the shell model in terms of a tensor interaction [22, 23]. However, the shell model is not an ideal tool to study single-particle energies, because it does not provide any description of the mean ?eld of the nucleus. Indeed, the single-particle states are theoretical constructs pertaining to the assumption that all nucleons move in a common, mean-?eld potential. Although such a mean-?eld picture of a nucleus gives its salient structural features, the actual nuclear state is certainly not a pure mean-?eld state. This can be understood in the spirit of the Kohn-Sham approach [24], in which mean-?eld orbitals represent correlated many-fermion states. The question of how much and what kind of correlations can be incorporated in a phenomenological EDF is still under debate. Certainly, the ?nal comparison with experimental data should not be done by looking at the single-particle energies themselves. It is rather unclear how to reliably extract these theoretical objects from measured properties, especially in open-shell systems. Rather one should attempt determining in theory these same observables that are measured in experiment, whereupon single-particle energies keep the meaning of auxiliary quantities that facilitate analyzing ?nal calculated results. In the present study, we follow this strategy by comparing measured and calculated masses of odd-A nuclei. The main focus of our analysis is on the spin-orbit (SO) properties of nuclei. The origin of the large nuclear one-body SO term, introduced in 1949 [3, 4], is still a matter of debate. On a microscopic level, a signi?cant part of the SO splitting comes from the two-body SO and tensor forces [25, 26, 27, 28], and there is also a signi?cant contribution from three-body forces [29, 30]. 4

     Suggestions to study tensor interactions within the self-consistent mean-?eld approach were made a long time ago [31, 32] but the scarce experimental data available (i.e., fairly short isotopic/isotonic chains) did not provide su?cient sensitivity to adjust the related coupling constants. On a one-body level, variation of the strength of the phenomenological SO potential due to the tensor force (sensitive to the e?ect of spin saturation) was studied in Refs. [33, 34]. Stimulated by the new data on neutron-rich nuclei, it is only very recently that the self-consistent treatment of the e?ective tensor

    interactions has been reintroduced [35, 36, 37]. Momentum-dependent zero-range tensor [31, 32] and spin-orbit (SO) [38] two-body interactions have the form ? VT e =

     1 t 2 e

     ?ä ? ??ä ? ? ? k ??S??k +k??S??k ,

     (2) (3) (4)

     ?ä ? ? ? VT o = to k ?? S ?? k, ? ? ??ä ? VSO = iW0 S ?? k ?Á k , where the vector and tensor spin operators read ? S = ?Ò1 + ?Ò2, ? Sij = 3 ?Ò i ?Ò j + ?Ò j ?Ò i ? ?Ä ij ?Ò 1 ?? ?Ò 2 . 1 2 1 2 2

     (5) (6)

     When averaged with one-body density matrices, these interactions contribute to the following terms in the energy-density functional (EDF) (see Refs. [39, 21] for derivations), HT = HSO =

     5 8 1 4

     te J n ?? J p + to (J 2 ? J n ?? J p ) , 0 3W0 J 0 ?? ??Ñ0 + W1 J 1 ?? ??Ñ1 ,

     (7) (8)

     where W1 = W0 and the conservation of time-reversal and spherical symmetries was assumed. Here, ?Ñt and J t are the neutron, proton, isoscalar, and isovector particle and vector SO densities [38, 39, 21] for t=n, p, 0, and 1, respectively. Within the EDF formalism, one extends the SO energy density (8) to the case of W1 = W0 [40]. Apart from the contribution of the SO energy density to the central potential, variation of the SO and tensor terms with respect to the densities yields the following form factors of the one-body SO potentials for neutrons and protons, W SO = n W SO p =

     5te +5to Jp + 8 5te +5to Jn + 8 5to Jn 4 5to Jp 4

     + +

     3W0 ?W1 +W ??Ñp + 3W04 1 ??Ñn , 4 3W0 ?W1 +W ??Ñn + 3W04 1 ??Ñp . 4

     (9) (10)

     Hence, it is clear that the only spectroscopic e?ect of tensor interaction is a modi?cation of the SO splitting of the single-particle levels. From the point of view of one-body properties, tensor interactions act similarly to two-body SO forces. However, the latter ones induce the SO splitting that is weakly dependent on shell ?lling. This is so because the corresponding form factors in Eqs. (9) and (10) are given by the radial derivatives of densities, ??Ñ = r d?Ñ , which are weakly dependent on shell e?ects. On r dr the other hand, the SO splitting induced by the tensor forces depends strongly on the shell ?lling, because the corresponding form factors are given by the SO densities, J = r J(r). Indeed, when only r one of the SO partners is occupied (spin-unsaturated system), the SO density J(r) is large, and when both partners are occupied (spin-saturated system), the SO

    density is small (see Ref. [35] for numerical examples). Within the SM, these simple SO e?ects are called ??level attraction?? or ??level repulsion?? [22, 23], but in reality they result from characteristic changes of the SO mean ?elds. Indeed, although the tensor SO mean ?elds are dominated by contributions from high-j orbitals, those of lower-j orbitals 5

     are also important. Instead of interpreting the e?ects of tensor interaction in terms of the ??level-level?? interactions, a more physical explanation can be given in terms of a two-step argument. First, the SO densities depend on occupations of the SO partners, and, second, the SO partners are split according to the one-body SO mean ?elds given by form factors (9,10).



     E (MeV)



     1g7/2 1h11/2 1h11/2 ? 1g7/2

     -10 50




     Neutron Number N

     Figure 3: Experimental di?erences of 11/2? and 7/2+ energies in the Z=51 isotopes E[1h11/2 ]?E[1g7/2 ] (diamonds) compared to di?erences E[1h11/2 ]?E[0+ ] (squares) and E[1g7/2 ]?E[0+ ] (circles) relative to the ground states of the Z=50 isotones. Open symbols show data for the Z=51 levels where there is no information from transfer reactions about the single-particle character of the states. (From Ref. [41].) In the present study, we apply tensor terms in the EDF to describe relative changes of positions of the proton 1h11/2 and 1g7/2 levels. Figure 3 shows experimental di?erences E[1h11/2 ] ? E[1g7/2 ] in the Z=51 isotopes between N=56 and 82 [41]. The ?gure also shows di?erences E[1h11/2 ] ? E[0+ ] and E[1g7/2 ] ? E[0+ ] (circles) relative to the ground states of the Z=50 isotones. In each case, the symbol E[state] denotes the total mass of the nucleus in the corresponding state. Figure 4 shows analogous results obtained within the spherical HFB calculations performed for the Skyrme interactions SLy4 [42] and SkO [16]. The HFB solutions in the Z=51 isotopes were obtained by self-consistently blocking one proton in the 1h11/2 and 1g7/2 orbits, while the standard unblocked calculations were performed in the Z=50 isotopes. Consequently, energy di?erences in Fig. 4 correspond to di?erences of the total calculated masses; thus they include the odd-particle polarization e?ects. Without tensor terms (top panels),

    E[1h11/2 ] ? E[1g7/2 ] values are almost constant with N. This illustrates the particle-number dependence of the standard SO splitting, which is dictated by gradients of densities. This result does not depend on the type of coupling which is di?erent for the two Skyrme interactions used. Indeed, for SLy4, one has W1 = W0 , and hence the proton SO splitting couples twice as strong to the gradient of the proton density than to the gradient of the neutron density (cf. Eq. (10)). On the other hand, for SkO, one has W1 ? ?W0 and the type of coupling is opposite ?C mostly to the gradient of the neutron density. Nevertheless, for both interactions, the standard SO splitting is almost constant. 6

     In the present study, we show results for tensor interaction to = 0 and te = 200 MeV fm5 , proposed in Ref. [35]. When such tensor interaction is taken into account, the results are qualitatively di?erent. Namely, di?erences E[1h11/2 ] ? E[1g7/2 ] are smallest near N=70 where the neutron SO densities are smallest. (The neutron 1g7/2 shell is almost ?lled while the neutron 1h11/2 shell is almost empty ?C i.e., the system is close to being spin-saturated.) On both sides of N=70, di?erences E[1h11/2 ] ? E[1g7/2 ] increase; for N < 70 (N < 70) this is due to decreasing (increasing) occupations of the 1g7/2 (1h11/2 ) shells. Therefore, on both sides, the neutron SO densities increase; hence, the 1h and 1g SO splittings decrease, resulting in the observed pattern of di?erences E[1h11/2 ] ? E[1g7/2 ].

     5 0 -5

     SO Wp

     HFB+SLy4 ~ 2??Ñp+??Ñn ??Ñ

     SO Wp

     HFB+SkO ~ 2??Ñn+??Ñp ??Ñ


     E (MeV)


     5 0 -5 -10 50

     HFB+SLy4 SO ?Wp ~ Jn

     HFB+SkO SO ?Wp ~ Jn




     1h11/2 1h11/2 - 1g7/2








     Neutron Number

     Figure 4: Di?erences of total calculated energies in the Z=51 isotopes, analogous to those shown in Fig. 3. Top panels show results obtained without tensor terms, to = te = 0, while bottom panels correspond to tensor-even terms of te = 200 MeV fm5 . In each case, we indicate the type of SO coupling SO SO within the proton standard SO (Wp ) and tensor SO (?Wp ) part of the form factor (10). Left and right panels show results obtained with the SLy4 [42] and SkO [16] interactions, respectively. The calculated pattern of the 1h11/2 ?C1g7/2 splitting re?ects the experimental trends; however, in experiment the minimum splitting is close to N=64, and the increase of splitting at N<64 is quite weak. In order to con?rm the observed pattern, it would be crucial to identify the proton 1h11/2 and 1g7/2 levels in the 101?105 Sb isotopes. Moreover, better Skyrme parametrizations are needed to shift the minimum from N=70 towards N=64 by shifting the neutron 3s1/2 and 2d3/2 levels up, closer to the 1h11/2 level. We note here in passing that in Ref. [36] such a shift was done arti?cially, without using the real self-consistent energies of the Gogny interaction.


     Particle Continuum and Shell Structure

     The many-body nuclear Hamiltonian does not describe just one nucleus (N, Z), but all nuclei that can exist. In this sense, a nucleus is never isolated (closed) but communicates with other nuclei through decays and captures. If the continuum space is not considered, this communication is not allowed: 7

     the system is closed. A consistent description of the interplay between scattering states, resonances, and bound states in the many-body wave function requires an OQS formulation (see Refs. [43] and references quoted therein). Properties of unbound states lying above the particle (or cluster) threshold directly impact the continuum structure. Coupling to the particle continuum is also important for weakly bound states, such as halos. A classic example of a threshold e?ect is the Thomas-Ehrman shift [44, 45] shown in Fig. 5 (left) which manifests itself in the striking asymmetry in the energy spectra between mirror nuclei having di?erent particle emission thresholds. As argued by Ikeda et al. [46],

     Thomas-Ehrman shift

     Alignment of the weakly bound state with the decay channel


     4946 3685 3089

     O+ n



     N+ p




     1/2+ 2365 1943


     ?Á C+




     7162 6049


     13 6C7

     13 7 N6

     ~ ( 0p)




     Figure 5: Examples of the continuum coupling on spectroscopic properties of nuclei. Left: the ThomasEhrman shift between 1/2+ and 3/2? states in mirror nuclei 13 N (one-proton threshold at 1943 keV) and 13 C (one-neutron threshold at 4946 keV). Right: the phenomenon of the alignment of the weakly bound state with the cluster decay channel; the appearance of the 0+ state in 16 O slightly below the 2 12 C+?Á threshold. clustering in nuclei becomes relevant for states in nuclei close to their cluster decay thresholds. This is a particular aspect of a general phenomenon of the alignment of near-threshold states with the decay channel which ?nds its explanation in generic features of the continuum coupling close to the threshold [43, 47, 48, 49]. The right-hand-side of Fig. 5 shows the example of the phenomenon of the alignment of the weakly bound 0+ state in 16 O at 6049 keV with the 12 C+?Á decay channel (threshold at 7162 keV). 2 Another splendid example is the 0+ resonance in 12 C at 7.65 MeV that lies slightly above the triple2 alpha threshold at 7.275 MeV. Both states, crucial for our understanding of stellar nucleosynthesis, have an extremely complex character in the language of standard SM. Their description requires proper treatment of many-body correlations and continuum. The mechanism of alignment of bound and unbound near-threshold states with the decay channel has its microscopic origin in the continuum coupling which is anomalously strong in the neighborhood of the particle-emission threshold [43]. The continuum-coupling energy correction to the CQS eigenvalue exhibits a cusp behavior [43, 48, 49] near the threshold which re?ects the change in the con?guration mixing due to the external coupling to the decay channel(s). This external coupling is responsible

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