Structure of 136 Sn and the Z = 50 magicity

The ﬁrst 2 + excited state in the neutron-rich tin isotope 136 Sn has been identiﬁed at 682(13) keV by measuring γ -rays in coincidence with the one proton removal channel from 137 Sb. This value is higher than those known for heavier even–even N = 86 isotones, indicating the Z = 50 shell closure. It compares well to the ﬁrst 2 + excited state of the lighter tin isotope 134 Sn, which may suggest that the seniority scheme also holds for 136 Sn. Our result conﬁrms the trend of lower 2 + excitation energies of even–even tin isotopes beyond N = 82 compared to the known values in between the two doubly magic nuclei 100 Sn and 132 Sn.


Introduction
Nuclei with a few valence particles or holes outside a doubly magic nucleus play an essential role in the study of nuclear structure. In particular, the properties of nuclei in the vicinity of 100 Sn and 132 Sn (Z = 50, N = 82) have attracted many studies in the past, as these nuclei are doubly magic while lying far away from the line of β stability. In addition, the long range of the Sn isotopic chain enables benchmark studies for the change of nuclear properties with greatly varying numbers of neutrons. It thus provides an opportunity to explore possible modifications in the shell structure beyond the two doubly magic nuclei toward the proton and neutron drip-lines. However, at present experimental knowledge for the tin isotopes heavier than 132 Sn is very limited and only exists up to 100 Sn on the proton-rich side [1].
For the neutron-rich tin isotopes, spectroscopic information beyond the 132 Sn core has been obtained experimentally only up to N = 84 [2][3][4]. The low-lying excited states in 133 Sn yield direct information on the single-particle states outside the N = 82 shell closure [2]. For 134 Sn, the energy of the first 2 + excited state (E x (2 + 1 )) was found to be much lower than that for 130 Sn [3], while the reduced transition probabilities between the 2 + 1 state and the ground state (B(E2)) are similar for those two nuclei [4]. This is very different from the case of heavier isotopic chains, such as Xe, Ba, and Ce, where B(E2) increases as E x (2 + 1 ) decreases [5]. It is therefore intriguing to extend the studies towards more neutron-rich Sn isotopes in order to investigate possible structure changes in this region.
The structural evolution in the Sn isotopes beyond N = 82 has been discussed theoretically with shell-model calculations employing different realistic effective interactions [6][7][8][9]. These shell-model calculations showed a very different trend in the energies of low-lying excited states as a function of the neutron number. Using effective nucleon-nucleon interactions, constant E x (2 + 1 ) values are predicted in Refs. [7,8], while Ref. [9] shows a decrease in the 2 + 1 excitation energy beyond N = 82 with an empirical interaction. Different E x (2 + 1 ) behaviors have also been suggested by mean-field approaches with different interactions [10][11][12]. Therefore, the E x (2 + 1 ) systematics are critical to understanding the structures in the Sn isotopes beyond N = 82. In the present work, we report on the first 2 + state in 136 Sn, which is the first spectroscopic study for this isotope with four valence neutrons in addition to the 132 Sn core.

Experiment
The experiment was carried out at the RIKEN Radioactive Isotope Beam Factory [13,14] operated by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo. The secondary beams including 137 Sb were produced by an in-flight fission reaction of a 238 U primary beam at 345 MeV/nucleon incident on a tungsten target. Secondary cocktail beams were selected and purified in the first stage of the BigRIPS fragment separator [15] by employing a wedge-shaped aluminum energy degrader with thickness of 0.8 g cm −2 at the dispersive focus. For a further purification, another 0.4 g cm −2 thick aluminum energy degrader was used in the second stage. Particle identification of the fission products was made event-by-event via the E − Bρ − TOF (time of flight) method as described in Ref. [16] using a similar setup.
The secondary beams impinged on a 1.1 g cm −2 thick 9 Be target to induce the secondary reactions. At the middle of the secondary target, the average energy of 137 Sb isotopes was around 240 MeV/nucleon. Reaction products were analyzed by the ZeroDegree spectrometer [15] and again identified using the E − Bρ − TOF method as for BigRIPS. The energy loss ( E) was measured by an ionization chamber located at the final focus to deduce the atomic number Z in combination with the time of flight, which was determined by two plastic scintillators placed at the entrance and the final focus of the ZeroDegree spectrometer. A trajectory reconstruction was made to determine the magnetic rigidity (Bρ) by position and angle measurements at the dispersive focus. With Bρ and TOF information, the mass-to-charge ratio A/Q was obtained. In addition, a LaBr 3 (Ce) scintillation detector located downstream of the ionization chamber was used for a total kinetic energy (TKE) measurement. The mass number (A) was obtained from the TKE-TOF correlation. Resolutions in Z , A/Q, and A for the Sn isotopes were 0.47 (FWHM-full width at half maximum), 6.5 × 10 −3 (FWHM), and 1.5 (FWHM), respectively. Figure 1(a) shows a two-dimensional plot of Z versus A/Q for the fragments produced from the 137 Sb secondary beams on the reaction target. The 136 Sn ions in the fully stripped (Q = Z ) charge state are contaminated by the 133 Sn 49+ ions in the hydrogen-like (Q = Z − 1) charge state as these two nuclei have similar A/Q values. Figure 1 after selecting the Sn isotopes by the Z gate of Z = 50.0 ± 0.4, as shown by the horizontal dashed lines in Fig. 1(a). The fully stripped 136 Sn fragments were unambiguously distinguished from the hydrogen-like 133 Sn 49+ contaminants in the plot. In order to detect the γ -rays emitted from decaying excited states, the secondary target was surrounded by the DALI2 spectrometer [17], which consisted of 186 NaI(Tl) scintillation detectors and covered the polar angles from 14 to 148 degrees with respect to the beam axis. The energy resolution and full energy peak efficiency were measured to be 9% (FWHM) and 22%, respectively, for the 0.662 MeV γ -rays emitted from a stationary 137 Cs source.

Results
The Doppler-shift-corrected γ -ray energy spectrum in coincidence with the 136 Sn isotopes produced via one proton knockout reactions from 137 Sb is shown in Fig. 2. A peak is clearly identified at 682(13) keV, which we attribute to the γ -ray decay from the 2 + 1 state to the ground state. The stated error of the energy includes statistical and systematic contributions. The latter is attributed to the uncertainties in the energy calibration (3 keV), and the ambiguities caused by the lifetime of the excited state as described in Ref. [18] causing an additional uncertainty of about 1% of the γ -ray energy. The experimental spectrum in Fig. 2 was fitted with a response function simulated with the GEANT4 [19] framework on top of an exponential background.

Discussion
In Fig. 3(a), the 2 + excitation energies for the even-even N = 86 isotones are displayed as a function of the proton number. The first 2 + excited state in 136 Sn is much higher than those for the neighboring nuclei, resulting in a concave pattern for the E x (2 + 1 ) systematics between Z = 50 and Z = 62. The high E x (2 + 1 ) value in 136 Sn indicates the presence of the Z = 50 magicity in the N = 86 isotones. The isotopic dependence of E x (2 + 1 ) for the even-even Sn isotopes is shown in Fig. 3(b). The newly measured E x (2 + 1 ) value for 136 Sn is close to that for 134 Sn at 725 keV [3].   along the isotopic chain beyond 132 Sn at 134 Sn and 136 Sn is consistent with the characteristic of the seniority scheme. Indeed, in the lighter Sn isotopes the 2 + 1 excited state stays almost constant in a very wide region between the doubly magic nuclei of 100 Sn and 132 Sn, which can be understood by a dominance of the seniority ν = 2 configuration [20,21], i.e., the excited states are created by breaking a neutron pair in the valence space. Our experimental finding of the similar E x (2 + 1 ) values for 134 Sn and 136 Sn may suggest that the seniority scheme also holds for the tin isotopic chain beyond N = 82 up to N = 86.
Another interesting feature of the E x (2 + 1 ) systematics along the tin isotopes is its asymmetric behavior between the two regions, one below N = 82 and the other above N = 82. The 2 + excitation energies in 134 Sn [3] and 136 Sn are about 500 keV lower than those values for the lighter isotopes  [6], the CWG (dot-dashed) interaction [7], the V low−k interaction (solid) [8], and the SMPN interaction (dotted) [9], are displayed for comparison. Panel (b) shows mean-field calculations using QRPA (dashed) [10], RQRPA (dot-dashed) [11], and the SHF method with paring forces DDDI (dotted) [12]. The vertical dotted line indicates the magic number N = 82. See text for details. between N = 50 and N = 82 [5]. The E x (2 + 1 ) asymmetric pattern might be due to a reduction of the pairing energy in the N > 82 region, because E x (2 + 1 ) is determined mostly by the strength of the pairing under the seniority scheme. Indeed, a quenching in the pairing gap is suggested from the decrease of the odd-even staggering of masses in the Sn isotopes across N = 82 in the recent mass measurement [22]. Although several reasons were proposed, such as a low level density [22] or the large average distance between the valence nucleons [7], the origin of the weakening of the neutron pairing is not clear. It is interesting to note that the asymmetric E x (2 + 1 ) pattern seen in the Sn isotopes with respect to N = 82 is not seen in their valence mirror nuclei with proton number around Z = 82 in the N = 126 isotones, where the protons occupy the same orbitals as the neutrons in the vicinity of 132 Sn. This fact implies that the weakening of the neutron pairing is caused by the structure change characteristic in this neutron-rich Sn region beyond N = 82.
The 2 + 1 excited states in the Sn isotopes beyond N = 82 have been studied by several shell-model calculations [6][7][8][9]. All these calculations, as shown in Fig. 4(a), use the neutron 2 f 7/2 , 3 p 3/2 , 1h 9/2 , 3 p 1/2 , 2 f 5/2 , 1i 13/2 single-particle orbitals between N = 82 and N = 126 and an inert 132 Sn core. Using the Hamiltonian obtained by reproducing the level scheme in 134 Sn [6], MCSM predicts a similar E x (2 + 1 ) value at 136 Sn to the one at 134 Sn and a slight increase in the 2 + 1 excitation energies for larger N . Calculations using different effective nucleon-nucleon interactions, denoted as CWG and V low−k , respectively, show a nearly constant E x (2 + 1 ). The CWG [7] and V low−k [8] interactions are derived from the CD-Bonn potential using the G matrix and constructing a low-momentum potential, respectively. The CWG calculations predict the dominance of the seniority ν = 2 configuration in these excited states [7]. By using an empirical interaction, the SMPN [9]  The E x (2 + 1 ) values for a wide range of the Sn isotopic chain have also been calculated by meanfield approaches [10][11][12], as shown in Fig. 4(b). The dashed line represents the QRPA calculations with the Skyrme force SLy4 [10], while the dot-dashed line exhibits the relativistic QRPA (RQRPA) calculations using the NL3 effective interaction and Gogny's paring forces (D1S) [11]. Both of these calculations predict similar E x (2 + 1 ) values for N = 84 and N = 86. The Skyrme-Hartree-Fock (SHF) model using a short-range paring force, namely, the density-dependent δ interaction (DDDI) [12], exhibits a steep decrease up to 138 Sn beyond N = 82. The RQRPA calculation shows an asymmetric pattern around N = 82; however, it overestimates the E x (2 + 1 ) values toward N = 82. The asymmetry is not reproduced by the QRPA and SHF calculations, although QRPA shows a good agreement with the lighter Sn isotopes. It seems that the two features, namely the constant E x (2 + ) values beyond N = 82 and the asymmetry around N = 82 as mentioned above, cannot be reproduced at the same time by these calculations.

Summary
In summary, the first 2 + excited state in 136 Sn has been identified for the first time by measuring γrays in conjunction with the one proton removal channel from 137 Sb. The measured E x (2 + 1 ) value of 682(13) keV in 136 Sn was found to be much higher than the neighboring N = 86 isotones, indicating a good Z = 50 shell closure. The E x (2 + 1 ) value stays almost constant in 134 Sn and 136 Sn, which suggests that the seniority scheme holds beyond N = 82 up to N = 86. The asymmetric pattern in the E x (2 + 1 ) systematics in the Sn isotopes above and below N = 82 might be due to the reduction of the pairing energy. Our result is fairly well produced by shell-model calculations while the mean-field approaches cannot reproduce the constancy of E x (2 + 1 ) and the asymmetry in the Sn isotopic chain at the same time. Further studies on the evolution of the 2 + 1 excitation energies in more neutron-rich tin isotopes with N > 86 are encouraged to see if the constancy in the 2 + 1 and higher excited states persists.