EPJ manuscript No. (will be inserted by the editor)
Shell-model structure of exotic nuclei beyond
A. Covelloa, L. Coraggio, A. Gargano, and N.Itaco
arXiv:0712.1715v2 [nucl-th] 14 Jan 2008
Dipartimento di Scienze Fisiche, Universit` di Napoli Federico II, and Istituto Nazionale di Fisica Nucleare, a Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy Received: date / Revised version: date Abstract. We report on a study of exotic nuclei around doubly magic 132 Sn in terms of the shell model employing a realistic e?ective interaction derived from the CD-Bonn
nucleon-nucleon potential. The shortrange repulsion of the bare potential is renormalized by constructing a smooth low-momentum potential, Vlow?k , that is used directly as input for the calculation of the e?ective interaction. In this paper we focus attention on the nuclei 134 Sn and 135 Sb which, with an N/Z ratio of 1.68 and 1.65, respectively, are at present the most exotic nuclei beyond 132 Sn for which information exists on excited states. Comparison shows that the calculated results for both nuclei are in very good agreement with the experimental data. We present our predictions of the hitherto unknown spectrum of 136 Sn. PACS. 21.60.Cs Shell model ?C 21.30.Fe Forces in hadronic systems and e?ective interactions ?C 27.60.+j 90 ?Ü A ?Ü 149
The study of neutron-rich nuclei around doubly magic 132 Sn is a subject of special interest, as it o?ers the opportunity to explore for possible changes in nuclear structure properties when moving toward the neutron drip line. In this context, great attention is currently focused on exotic nuclei beyond the N = 82 shell closure. This is motivated by the fact that some of the data that have become available appear to be at variance with what one might expect by extrapolating the existing results for N < 82 nuclei. In particular, some peculiar properties have been recently observed in the two nuclei 134 Sn and 135 Sb which, with an N/Z ratio of 1.68 and 1.65, respectively, are at present the most exotic nuclei beyond 132 Sn for which information exists on excited states. This is the case of the ?rst 2+ state in 134 Sn which, lying at 726 keV excitation energy, is the lowest ?rst-excited 2+ level observed in a semi-magic eveneven nucleus over the whole chart of nuclides. As for 135 Sb, there is a signi?cant drop in the energy of the lowest-lying 5/2+ state as compared to the values observed for
the Sb isotopes with N ?Ü 82. These data might be seen as the onset of a shell-structure modi?cation which, starting at N = 83 ? 84, is expected to produce more evident e?ects for larger neutron excess. From this viewpoint, the anomalously low position of the 5/2+ state in 135 Sb may be attributed to a downshift of the d5/2 proton level relative to the g7/2 one caused by a more di?use nuclear surface produced by the two neutrons beyond the 82 shell closure. Actually, a good description of 135 Sb has been obtained in the
shell-model study of refs. [1,2] using experimental singleparticle (SP) energies with this downshift set at 300 keV. However, while the same kind of calculation also provides a satisfactory agreement with experiment for 134 Sn , this is not the case for the one proton, one neutron nucleus 134 Sb . In recent work [4,5,6] we have shown that the properties of these three nuclei are well accounted for by a unique shell-model Hamiltonian with SP energies taken from experiment and the two-body e?ective interaction derived from the CD-Bonn nucleon-nucleon (N N ) potential . The short-range repulsion of the latter has been renormalized by use of the low-momentum potential Vlow?k , as we shall brie?y discuss in sect. 2.
In this paper, we focus attention on the two nuclei Sn and 135 Sb. Based on the good agreement between our results and the available experimental data obtained in [4,6], we ?nd it interesting to report here the complete calculated low-energy spectra of these two nuclei. We hope that this may stimulate, and be helpful to, future experiments on these highly important nuclei. With the same motivation, we also present our predictions of the hitherto unknown spectrum of 136 Sn with four neutrons outside 132 Sn, which makes an N/Z ratio of 1.72. The outline of the paper is as follows. In sect. 2 we give a brief description of the theoretical framework in which our realistic shell-model calculations have been performed. In sect. 3 we give some details of the calculations and present our results together with the experimental data available for 134 Sn and 135 Sb. Some concluding remarks are given in sect. 4.
A. Covello et al.: Shell-model structure of exotic nuclei beyond
2 Outline of theoretical framework
The starting point of any realistic shell-model calculation is the free N N potential. There are, however, several highquality potentials, such as Nijmegen I and Nijmegen II , Argonne V18 , and CD-Bonn
, which ?t equally well (?Ö2 /datum ?Ö 1) the N N scattering data up to the inelastic threshold. This means that their on-shell properties are essentially identical, namely they are phase-shift equivalent. In our shell-model calculations we have derived the e?ective interaction from the CD-Bonn potential. This may raise the question of how much our results may depend on this choice of the N N potential. We shall comment on this point later in connection with the Vlow?k approach to the renormalization of the bare N N potential. The shell-model e?ective interaction Ve? is de?ned, as usual, in the following way. In principle, one should solve a nuclear many-body Schr??dinger equation of the form o H??i = Ei ??i , (1)
3 Calculations and results
In our calculations we assume that 132 Sn is a closed core and let the valence neutrons occupy the six levels 0h9/2 , 1f7/2 , 1f5/2 , 2p3/2 , 2p1/2 , and 0i13/2 of the 82-126 shell, while for the odd proton in 135 Sb the model space includes the ?ve levels 0g7/2 , 1d5/2 , 1d3/2 , 2s1/2 , and 0h11/2 of the 50-82 shell. As mentioned in the previous section, the two-body matrix elements of the e?ective interaction are derived from the CD-Bonn N N potential renormalized through the Vlow?k procedure with a cuto? momentum ?? = 2.2 fm?1 . This value of ?? is in accord with the criterion given in ref. . The computation of the diagrams ? included in the Q-box is performed within the
harmonicoscillator basis using intermediate states composed of all possible hole states and particle states restricted to the ?ve shells above the Fermi surface. The oscillator parameter used is ?? ?Ø = 7.88 MeV. h As regards the SP energies, they have been taken from experiment. In particular, the spectra  of 133 Sb and 133 Sn have been used to ?x the proton and neutron SP energies, respectively. The only exceptions are the proton ?s1/2 and neutron ?i13/2 , whose corresponding levels are still missing. Their values have been taken from refs.  and , respectively, where it is discussed how they are determined. All the adopted values are reported in ref. .
with H = T + VN N , where T denotes the kinetic energy. This full-space many-body problem is reduced to a smaller model-space problem of the form P He? P ??i = P (H0 + Ve? )P ??i = Ei P ??i . (2)
Here H0 = T + U is the unperturbed Hamiltonian, U being an auxiliary potential introduced to de?ne a convenient single-particle basis, and P denotes the projection operator onto the chosen model space. As pointed out in the Introduction, we ??smooth out?? the strong repulsive core contained in the bare N N potential VN N by constructing a low-momentum potential Vlow?k . This is achieved by integrating out the high-momentum modes of VN N down to a cuto? momentum ??. This integration is carried out with the requirement that the deuteron binding energy and phase shifts of VN N up to ?? are preserved by Vlow?k . A detailed description
of the derivation of Vlow?k from VN N as well as a discussion of its main features can be found in refs. [8,11]. However, we should mention here that shell-model e?ective interactions derived from di?erent phase-shift equivalent N N potentials through the Vlow?k approach lead to very similar results . In other words, Vlow?k gives an approximately unique representation of the N N potential. Once the Vlow?k is obtained, we use it as input interaction for the calculation of the matrix elements of the shell-model e?ective interaction. The latter is derived by employing a folded-diagram method, which was previously applied to many nuclei using G-matrix interactions . Since Vlow?k is already a smooth potential, it is no longer necessary to calculate the G matrix. We therefore perform shell-model calculations following the same procedure as described, for instance, in [13,14], except that the G matrix used there is replaced by Vlow?k . More precisely, we ? ?rst calculate the so-called Q-box  including diagrams up to second order in the two-body interaction. The shellmodel e?ective interaction is then obtained by summing ? up the Q-box folded diagram series using the Lee-Suzuki iteration method .
?? ????o D
Fig. 1. Experimental and calculated spectrum of
Let us start with 134 Sn. From ?g. 1 we see that while the theory reproduces very well all the observed levels [20, 21], it also predicts, in between the 6+ and 8+ states, the existence of ?ve states with spin ?Ü 5. Clearly, the latter could not be seen from the ?Ã-decay of the 8+ state
A. Covello et al.: Shell-model structure of exotic nuclei beyond
populated in the spontaneous ?ssion experiment of ref. . Very recently, the B(E2; 0+ ?ú 2+ ) value in 134 Sn has 1 been measured  using Coulomb excitation of neutronrich radioactive ion beams. We have calculated this B(E2) with an e?ective neutron charge of 0.70 e, according to our early study . We obtain B(E2; 0+ ?ú 2+ ) = 0.033 1 e2 b2 , in excellent agreement with the experimental value 0.029(4) e2 b2 .
Table 1. Experimental and calculated excitation energies (in MeV) for 135 Sb Calc. J E
?Ð 7+ 2 5+ 2 3+ 2 1+ 2 11 + 2 9+ 2 5+ 2 7+ 2 7+ 2 9+ 2 15 + 2 9+ 2 5+ 2 5+ 2 13 + 2 7+ 2 3+ 2 19 + 2
proton interaction, we have found that the 5/2+ state is of admixed nature. We have calculated the B(M 1; 5/2+ ?ú 7/2+) making use of an e?ective M 1 operator which includes ?rst-order diagrams in Vlow?k . Our predicted value is 4.0 ?? 10?3 ?Ì2 . Keeping in mind that in our calculaN tion we do not include any meson-exchange correction, the agreement between the experimental and calculated B(M 1) may be considered quite satisfactory.
Table 2. Calculated excitation energies (in MeV) for J?Ð 0 2+ 4+ 6+ 4+ 2+ 2+ 5+ 4+ 3+ 8+
Calc. 0.0 0.715 1.020 1.163 1.170 1.329 1.417 1.465 1.660 1.724 1.811
Expt. J E
?Ð 7+ 2 5+ 2 3+ 2 11 + 2 9+ 2
0.0 0.391 0.509 0.678 0.750 0.813 0.924 0.938 1.031 1.108 1.124 1.144 1.146 1.182 1.209 1.231 1.263 1.268
0.0 0.282 0.440 0.707 0.798
7+ 2 9+ 2 15 + 2
1.014 1.027 1.117
7+ 2 19 + 2
In table 2 we report the calculated excitation energies of 136 Sn up to about 1.8 MeV. We ?nd that three lowest states, 2+ , 4+ , and 6+ , lie at practically the same energy as in 134 Sn. Above the 6+ level there are seven states in an energy interval of about 650 keV. This pattern is quite di?erent from that predicted for the spectrum of 134 Sn, where a rather pronounced gap (about 0.5 MeV wide) exists between the 6+ state and the next excited state with J ?Ð = 2+ .
The calculated and observed levels [3,17] of 135 Sb up to about 1.3
MeV excitation energy are reported in table 1. We see that the agreement between theory and experiment is very good. It is worth noting that our calculation reproduces the observed 5/2+ state within 100 keV. This is a relevant result as it argues against the suggestion  that the low position of this state, assumed to be essentially of single-particle character, is related to a relative shift of the proton d5/2 and g7/2 orbits induced by the neutron excess. From table 1 it can be seen that we predict several low-lying excited states which have no experimental counterpart. We hope that these predictions will be veri?ed by further experimental work. In the very recent work of refs. [23,24] the lifetime of the 5/2+ state in 135 Sb has been measured. A very small upper limit for the B(M 1), 0.29??10?3 ?Ì2 , was found, thus N evidencing a strongly hindered transition, which may be seen  as a con?rmation of the single-particle nature of the 5/2+ state. This is at variance with the outcome of our calculation. In fact, as a consequence of the neutronWe have presented here the results of a shell-model study of neutron-rich nuclei around 132 Sn, focusing attention on 134 Sn and 135 Sb which are at present the most exotic nuclei with valence neutrons beyond N = 82. The two-body ? e?ective interaction has been derived by means of a Q-box folded-diagrams method from the CD-Bonn N N potential, the short-range repulsion of the latter being renormalized by use of the low-momentum potential Vlow?k . Our results for both nuclei are in very good agreement with the observed spectroscopic properties. These results, considered along with those obtained for 134 Sb , show that to explain the presently available data on neutron-rich nuclei beyond 132 Sn there is no need to invoke shell-structure modi?cations. These data, however, are still rather scanty and we have found it challenging to make predictions which may stimulate further experimental e?orts to study this kind of exotic nuclei. In particular, we hope that our predictions for the hitherto unknown 136 Sn will be veri?ed in a not too distant future.
A. Covello et al.: Shell-model structure of exotic nuclei beyond
This work was supported in part by the Italian Ministero dell??Istruzione, dell??Universit` e della Ricerca (MIUR). a
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