Isomerism in the"south-east"of $^{132}$Sn and a predicted neutron-decaying isomer in $^{129}$Pd

Excited states in neutron-rich nuclei located south-east of $^{132}$Sn are investigated by shell-model calculations. A new shell-model Hamiltonian is constructed for the present study. The proton-proton and neutron-neutron interactions of the Hamiltonian are obtained through the existing CD-Bonn $G$ matrix results, while the proton-neutron interaction across two major shells is derived from the monopole based universal interaction plus the M3Y spin-orbit force. The present Hamiltonian can reproduce well the experimental data available in this region, including one-neutron separation energies, level energies and the experimental $B(E2)$ values of isomers in $^{134,136,138}$Sn, $^{130}$Cd, and $^{128}$Pd. New isomers are predicted in this region, $e.g.$ in $^{135}$Sn, $^{131}$Cd, $^{129}$Pd, $^{132,134}$In and $^{130}$Ag, in which almost no excited states are known experimentally yet. In the odd-odd $^{132,134}$In and $^{130}$Ag, the predicted very long $E2$ life-times of the low-lying $5^{-}$ states are discussed, demanding more information on the related proton-neutron interaction. The low-lying states of $^{132}$In are discussed in connection with the recently observed $\gamma$ rays. The predicted $19/2^{-}$ isomer in $^{129}$Pd could decay by both electromagnetic transitions and neutron emission with comparable partial life-times, making it a good candidate for neutron radioactivity, a decay mode which is yet to be discovered.


Introduction
On the journey towards the neutron drip-line one needs a reliable theoretical model which incorporates the known features of the nuclear many-body system and has enough predictive power for a range of unexplored nuclei. The nuclear shell model is one such, providing the basic framework for understanding the detailed structure of complex nuclei as arising from the individual motion of nucleons and the effective nuclear interactions between them. In the shell model, doubly magic nuclei, especially those far from the line of stability, such as 132 Sn, act as cornerstones for exploring the unknown regions.
Experimentally, the observation of isomers has been key to the understanding of the shell structure and the development of the shell model [1]. Recently, nuclei around 132 Sn have been the subject of intensive experimental studies with respect to the persistence of the N = 82 shell gap and its relevance to the astrophysical r-process path. Early β-decay results seemed to indicate a substantial shell quenching [2], while isomeric spectroscopy studies gave evidence for the persistence of the N = 82 shell down to Z = 46, 128 Pd [3,4,5]. Along the Z = 50 line isomeric states were also observed in 134,136,138 Sn [6,7], and mixing between seniority-2 and -4 configurations was revealed * yuancx@mail.sysu.edu.cn * * liuzhong@impcas.ac.cn for the 6 + isomer of 136 Sn [7]. Furthermore, mass measurements have been crucial for experimental determination of the shell gaps [8,9]. In addition, isomers in the region far from the β-stability line could serve as stepping stones towards the drip-lines. For example, the high-spin (19/2 − ) isomer in 53 Co provided the first example of proton radioactivity [10]. Similarly in the very neutron-rich region, a neutron may "drip" from an isomer before the neutron drip-line itself is reached, if the isomer's excitation energy takes it above the neutron separation energy [11,12].
In this paper, shell-model calculations are performed to investigate the isomerism in the south-east quadrant of 132 Sn, i.e. with Z ≤ 50 and N ≥ 82, including the possibility of neutron radioactivity from such isomers.

Effective Hamiltonian
The construction of an effective Hamiltonian is one of the key elements in a shell-model study. The model space for the present work is π0 f 5/2 , π1p 3/2 , π1p 1/2 , π0g 9/2 , and ν1 f 7/2 , ν2p 3/2 , ν2p 1/2 , ν0h 9/2 , ν1 f 5/2 , ν0i 13/2 , corresponding to the Z = 28 − 50 and N = 82 − 126 major shells, respectively. Below 132 Sn, the robustness of the N = 82 shell gap has been experimentally examined and confirmed down to Cd and Pd [3,5,8,9]. The core excited states in 131 In are found to be at nearly 4 MeV [4], thus this model space is suitable for the investigation of the low-lying states around or lower than 2 MeV in Sn, In, Cd, Ag, and Pd isotopes with N > 82. So far there is no well established effective Hamiltonian for this model space due to the lack of experimental data on the excited states in this most neutron-rich region around 132 Sn. An effective Hamiltonian for this model space is proposed below. In a very recent work, shell-model calculations for 132 In were performed employing a modern realistic effective interaction and two-body matrix elements deduced from the 208 Pb region [13].
The single-particle energies for the four proton orbits and the six neutron orbits in the present model space are fitted to the reported energies of the single-particle states of 131 In and 133 Sn, respectively. These energies are from Ref. [14] and the recently discovered π1p 1/2 and π1p 3/2 single-hole states in 131 In [15,16]. The single-particle energy for the ν0i 13/2 orbit in Ref. [14] is estimated from the excitation energy of the 10 + state in 134 Sb [17]. The present Hamiltonian fixes the relative single-particle energies to the observed excited states. It is reasonable as the present work concentrates on the excitation energies of levels and neutron separation energies.
The proton-proton interaction is based on the proton-proton part of jj45pna Hamiltonian, which has been derived from the CD-Bonn potential through the G matrix renormalization method by Hjorth-Jensen and is included in the OXBASH package [18]. The theoretical method to derive the jj45pna effective interaction and its application in the A ∼ 100 region is described in Ref. [19]. Recently, the Hamiltonian jj45pna was also used to investigate the β decay of 113 Cd and 115 In [20], and the lowlying states of In isotopes around A=125 [21]. The strength of this interaction is modified by a factor 0.74 to reproduce the low-lying states of 130 Cd. The neutron-neutron interaction is from the neutron-neutron part of CWG Hamiltonian, which is derived from the CD-Bonn renormalized G matrix and used to study nuclei around 132 Sn [22].
The proton-neutron interaction is calculated through an effective nucleon-nucleon, monopole-based universal interaction V MU [23] plus a spin-orbit force from M3Y [24](V MU +LS). The validity of the V MU +LS interaction in shell-model calculations has been examined in different regions of the chart of nuclei. The structure features of neutron rich C, N, O [25], Si, S, Ar, Ca [26], Cr, and Fe isotopes [27] have been nicely described by shell-model calculations by taking V MU +LS as the proton-neutron interaction between the p proton shell and the sd neutron shell [25], the sd proton shell and the p f neutron shell [26], and the p f proton shell and the gds neutron shell [27], respectively. For example, the neutron drip-lines for carbon, nitrogen and oxygen isotopes are simultaneously explained, revealing the impact of the proton-neutron interaction on the evolution of the nuclear shell [25]. Recently, the first 4 + state of 44 S has been identified as a high-K isomer [28] through the Hamiltonian suggested in Ref. [26]. In the heavier region, close to 132 Sn, the change of the energy difference between the 10 + and 7 − yrast levels in the N = 80 isotones down to 126 Pd is well explained by V MU +LS [29]. Thus it is natural and reasonable to use this interaction as the proton-neutron interaction in the present study. In the present Hamiltonian the strength of the central-force parameters of V MU is enhanced 1.07 times the original one in Ref. [23] to give a better description of the one-neutron separation energies S n . It should be noted that the original form of V MU comes from the effective Hamiltonian in the sd and p f regions. In psd region, the strength of its central part is reduced by a factor of 0.85 to reproduce the binding energies of the B, C, N, and O isotopes. The new Hamiltonian is named as j j46 in the following discussion as it includes 4 and 6 valence proton and neutron orbits, respectively. The present shell-model calculations are performed using the code OXBASH [18].
It should be mentioned that the present Hamiltonian operates in the particle-particle model space. The doubly magic nucleus 132 Sn has fully occupied valence protons and no valence neutrons in the present model space. Starting from 132 Sn, the proton-hole energies of 131 In and the neutron-particle energies of 133 Sn are not directly taken as the single particle energies in the Hamiltonian, but are modified by the residual proton-proton and proton-neutron interactions, respectively. The proton-hole states in the present work are affected by the missing correlations due to the removal of protons from the fully occupied 28-50 shell. In the following discussion, the configurations are written in the proton-hole neutron-particle scheme for simplicity.

Results and Discussion
With the j j46 Hamiltonian described above, the properties of 133  separation energies [30] in this region nicely. Both the singleparticle energy of the ν1 f 7/2 orbit and the proton-neutron interactions involving the fully occupied proton orbits contribute to S ( j j46) n of 133 Sn. Its value together with the other observed S n values are used to constrain the strength of the proton-neutron interaction in the present Hamiltonian as discussed in the previous section.
The levels of neutron-rich Sn isotopes are presented in Fig. 1. The present Hamiltonian reproduces the known low-lying states of 134,136,138 Sn well, especially the positions of the 2 + states in 134,136,138 Sn and the increasing trend of the energies of 4 + and 6 + states from 134 Sn to 138 Sn. The dominant configurations of the 8 + states in 134 Sn and 138 Sn are 99.55% ν(1 f 7/2 )(1h 9/2 ) and 41.83% ν(1 f 7/2 ) 5 (1h 9/2 ), respectively. This state in 136 Sn is dominated by the ν(1 f 7/2 ) 4 configuration (84.34%). If all the yrast 6 + and 4 + levels in 134,136,138 Sn were dominated by a pure seniority-2 configuration, the B(E2; 6 + → 4 + ) value in 136 Sn would be expected to be the lowest among these three isotopes, but their experimental results are decreasing from 134 Sn to 138 Sn. The seniority scheme of the low-lying states in these three isotopes can be discussed through the comparison between the experimental results and the shell model calculations. The observed B(E2; 6 + → 4 + ) value indicates a mixing of seniority-2 and -4 configurations in the 4 + state of 136 Sn by comparing the results from a realistic effective interaction and the empirical modification of ν(1 f 7/2 ) 2 matrix elements [7,33]. The present calculation also gives the decreasing B(E2) values between 6 + and 4 + states from 134 Sn to 138 Sn, which will be shown later.
The large energy differences between the 6 + and 8 + states in these three nuclei suggest that the 8 + states are not isomeric. However the small energy difference between the 17/2 − and 21/2 − states in 135 Sn implies a 21/2 − metastable state. Details for this possible isomer in 135 Sn will be given later. However, no such isomer is predicted in 137 Sn (not shown in Fig. 1). The first 21/2 − state of 135 Sn is dominated by a ν(1 f 7/2 ) 2 (0h 9/2 ) configuration (96.7%). Compared with 135 Sn the first 21/2 − level in 137 Sn is expected to be more mixed because of the increasing number of valence neutrons and/or the change of the shell structure. Fig. 2 presents the effective single-particle energies (ESPE) of the neutron orbits in Sn isotopes. ESPE are defined as [31], where ε core j is the single-particle energy relative to the core, ψ| N j ′ |ψ is the shell-model occupancy of the j ′ orbit and V j j ′ is the monopole part of the two-body interaction. As shown in Fig. 2, the single particle energies do not change much in the Sn isotopes. Therefore the main difference between the 21/2 − states in 135 Sn and 137 Sn arises from the two additional valence neutrons in 137 Sn. The configuration of the first 21/2 − level in 137 Sn is a mixture of ν(1 f 7/2 ) 4 (0h 9/2 ) (61.3%), ν(1 f 7/2 ) 2 (2p 3/2 ) 2 (0h 9/2 ) (9.97%), ν(1 f 7/2 ) 3 (2p 3/2 )(0h 9/2 ) (9.05%), and ν(1 f 7/2 ) 2 (1 f 5/2 ) 2 (0h 9/2 ) (3.38%), very different from the almost pure ν(1 f 7/2 ) 2 ν(0h 9/2 ) configuration in 135 Sn. Levels of the 132,133 In, 130,131 Cd, and 128,129 Pd isotopes are presented in Fig. 3. Some of them are possibly isomers. The ground state of 132 In is found to be 7 − with a configuration of π(0g 9/2 ) −1 ν(1 f 7/2 ), in agreement with the experimental assignment [14]. Our calculations indicate a very low 5 − state in 132 In, which can be a candidate for an isomer. With two more neutrons and two more proton holes respectively, 134 In and 130 Ag have similar structure, 7 − for ground state and very low 5 − for first excited state. These results depend on the details of the proton-neutron interaction between π(0g 9/2 ) −1 and ν(1 f 7/2 ) orbits, for which the experimental information is still rare.
Recently six γ rays were observed following the β-delayed neutron emission from 133 Cd and assumed to be emitted from the excited states of 132 In [13]. Due to the low statistics and    the lack of γ-γ coincidence, it was difficult to establish a level scheme with those observed γ rays [13]. The spin parity of the ground state of 133 Cd is 7/2 − from Ref. [14] and the present shell-model calculation. As the β decay energy (∼ 13 MeV) of 133 Cd is much larger than the neutron separation energy (∼ 3 MeV) of 133 In, the ground state of 133 Cd can decay to many high excited states beyond the neutron emission threshold of 133 In, with spin parity 5/2 − to 9/2 − through Gamow-Teller transitions and 3/2 + to 11/2 + through first forbidden transitions. Thus many states in 132 In could be populated in the β-delayed neutron emission of 133 Cd. The low-lying states of the π −1 ν configurations in 132 In calculated by the present Hamiltonian are shown in Fig. 4. The present shell-model results are similar to those shown in Fig. 4(a) of Ref. [13], especially for the π(1p 1/2 ) −1 ν(1 f 7/2 ), π(0g 9/2 ) −1 ν(2p 3/2 ), and π(1p 3/2 ) −1 ν(1 f 7/2 ) multiplets, though the proton-neutron interaction is calculated through V MU +LS in this work, while it is derived from the CD-Bonn potential with V low−k approach in [13].
In both calculations the 7 − state is the lowest and 1 − the highest among the π(0g 9/2 ) −1 ν(1 f 7/2 ) multiplets, however the other states are slightly different. The present work predicts that 5 − is first excited state and a little lower than 6 − state, while the shell-model calculation in Ref. [13] gave opposite prediction. Similarly the present shell-model results may also explain most of the observed γ rays in 132 In [13]. Because of the uncertainty of the shell-model calculations and extra complexity introduced by the low-lying π(1p 1/2 ) −1 ν(1 f 7/2 ) multiplets, more experimental information on the excited states of 132 In, 134 In and 130 Ag are highly desired and crucial in understanding the proton-neutron interactions in this region.
All the seniority isomers experimentally observed in 134,136,138 Sn, 130 Cd and 128 Pd are well reproduced. Their semimagic nature validates the neutron-neutron and proton-proton parts of the shell-model interactions in the present work. In some other nuclei in this region, some of the levels are possibly isomeric because of the slow transition rates resulting from low transition energies. These isomer candidates are listed in Table 2 together with those experimentally confirmed in Sn isotopes and N=82 isotones. τ E2 is the mean life-time of E2 transitions obtained from the theoretical reduced transition probability B(E2) values and predicted transition energies. The experimental B(E2) expt values of 134,136,138 Sn are from Ref. [7]. The two B(E2) expt values of 130 Cd are due to the two possible decay transition energies [3]. The effective charges for calculation of B(E2) values for protons and neutrons are e p = 1.7e and e n = 0.7e, respectively, which are similar to those used in Ref. [7].
The 21/2 − isomer in 135 Sn is analogous to the 6 + isomer in 134 Sn, but its B(E2) value is much larger than that of 134 Sn, as shown in Table. 2. The one-body transition density of the neutron 1 f 7/2 orbit from 6 + to 4 + in 134 Sn is almost the same as that from 21/2 − to 17/2 − in 135 Sn, and the B(E2) enhancement in 135 Sn is mainly due to the transition from 0h 9/2 to 1 f 5/2 and 1 f 7/2 to 1p 3/2 . Although the occupancies of 1 f 5/2 and 1p 3/2 in the 17/2 − state are small, the large number of 0h 9/2 and 1 f 7/2 particles in the 21/2 − state result in a large one-body transition density. The transition energy between 21/2 − and 17/2 − in 135 Sn is predicted to be almost the same as that between 6 + and 4 + in 134 Sn. τ E2 around 100 ns is predicted for the 21/2 − isomer in 135 Sn.
The yrast 5 − states in 132,134 In and 130 Ag are predicted to be closely below the 6 − levels and only around 70 keV above the 7 − ground states, as shown in Fig. 4 for 132 In, leading to long life-times. A much larger τ E2 in 132 In is predicted because of the small B(E2) value caused by the cancellation between the proton and neutron contributions. It should be noted that due to the lack of experimental information on the proton-neutron interaction in this quadrant of 132 Sn, large uncertainties related to the isomerism in these odd-odd nuclei are not unexpected in the present shell-model calculations. For example, the 5 − isomer will disappear if it is higher than the 6 − state. Or alternatively the 4 − state could be isomeric, if it lies below the 5 − state.
19/2 − isomers are predicted in the N = 83 isotones 131 Cd and 129 Pd, with τ E2 ∼ 100 ns and ∼ 1 µs, respectively. The excitation energy of the predicted 19/2 − isomer in 129 Pd is around 0.4 MeV above its neutron separation energies predicted by the present work and by Moller et al. [32] (Table 1) and so it may decay to the ground state of 128 Pd by emitting a neutron with an orbital angular momentum of l = 9 (not to 2 + state of 128 Pd because of its 1.311 MeV excitation energy). The half-life of neutron emission is calculated by using the widely-used formula of the two-potential approach [35], in which both the preexponential factor and exponential factor are explicitly defined. The potential that the emitted neutron feels is a sum of the nuclear potential, the spin-orbit potential and the centrifugl potential. The form of both the Woods-Saxon nuclear potential and the spin-orbit potential and the parameters used are taken from textbook [36], l(l + 1) where µ is the reduced mass of the neutron, Q n is the decay energy, r 1 and r 2 are classical turning points defined by Q n = V + 2 2µ l(l+1) r 2 , and G is the global quantum number. The predicted life-time for neutron emission from the 19/2 − state of 129 Pd to the ground state of 128 Pd ranges from 0.3 to 10 µs with the decay energy Q n = 0.35-0.50 MeV, comparable to that of the calculated electromagnetic decay. The global quantum number in the Bohr-Sommerfeld condition is fixed at 9 due to the very low decay energy.

Summary
In summary, a shell-model study has been performed in the south-east of 132 Sn to search for possible isomeric states. A new shell-model Hamiltonian has been constructed for the present investigation. The proton-proton and neutron-neutron interactions, which are each limited to one major shell, have been obtained from existing CD-Bonn G matrix calculations. The proton-neutron interaction across two major shells is calculated through the V MU plus M3Y spin-orbit interaction. The present Hamiltonian, j j46, is able to reproduce well the one-neutron separation energies, level energies, and B(E2) values of the already observed isomers in this region. New isomeric states are predicted and their structures are discussed. The predicted 19/2 − isomer in 129 Pd could be a candidate for neutron radioactivity.