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The analysis of neutron-rich isotopes near the doubly magic nucleus
132 Sn can provide fascinating findings related to nuclear physics and nuclear astrophysics. The abundance peak appears atA∼130 , which is formed through the rapid neutron capture process [1, 2]. The properties of doubly magic132 Sn have been explored and confirmed in both experiments and theories [3−8]. In this nuclei region, tellurium isotopes have attracted experimental and theoretical research interest [5, 9−13]. For example, the g-factor of134 Te in the 4+ state was measured, which provides direct insight into the single particle structure [9]. The state (17+ ) of132 Te was observed to be 6.166 MeV using the reaction9 Be(238 U, f ) with a beam energy of 6.2 MeV/u at GANIL [14]. A significant energy difference exists in this state between experiments and shell-model calculations, which should conceal unknown information about the nuclear structure.In theory, the extended pairing plus multipole-multipole force and monopole correction terms model (EPQQM) provides a suitable method to describe both the low-lying states and core excitations [15−18]. For example, the ordering and energies of the low-lying isomers in
129 Cd are predicted and determined by using the implemented phase-imaging ion-cyclotron-resonance method [18, 19]. The 16+ level in128 Cd is predicted as a spin-trap isomer feeding the known 16+ of128 In throughβ− decay [20]. In addition to monopole interactions, it is necessary to consider the cross-shell excitations to study the properties of these neutron-rich nuclei in this region. For example, identifying the isomer state of level 19/2+ at 1942 keV in133 Ba requires the cross-shell orbits lying above the energy gapN=82 . As reported in Ref. [21], the interactions without core excitations cannot provide the B(E1) value for the transition fromJπ=19/2+ toJπ=19/2− .This model has an advantage for studying monopole effects by employing monopole correction (Mc) terms. For example, in the southwest quadrant (
Z≤50,N≤82 ) of132 Sn, the level spectra and the energy gap acrossN=82 can be modified by monopole correction between neutron orbith11/2 andf7/2 [15]. In the northeast quadrant (Z≥50,N≥82 ), five monopole terms are used to describe core excitations and high-spin levels, and the states 2− and 9− in136,138 Te are predicted as a spin-trap structure coupled by the neutron intruder orbiti13/2 [22]. Different effects of tensor forces are also discussed together with the monopole-driven shell evolutions [23], as well as other ones in the nucleon-nucleon interaction [24−27]. For the particle-hole nuclei in the northwest region of132 Sn, a suitable interaction has been found, and the spectra of Sb and Te isotopes are well described as single-orbital couplings and cross-shell excitations [18, 28]. The transition probabilities in these nuclei are also calculated and reproduced well through comparisons with the known data.In this study, we investigate high-spin levels and monopole effects in particle-hole nuclei near
132 Sn by employing the interaction in Ref. [18]. The shell-model code NUSHELLX@MSU is used for the calculations [29]. -
In this work, we use the Hamiltonian in the proton-neutron (
pn ) representation [18]:H=Hsp+HP0+HP2+HQQ+HOO+HHH+Hmc=∑α,iεiac†α,icα,i−12∑J=0,2∑ii′gJ,ii′∑MP†JM,ii′PJM,ii′−12∑ii′χ2,ii′b4∑M:Q†2M,ii′Q2M,ii′:−12∑ii′χ3,ii′b6∑M:O†3M,ii′O3M,ii′:−12∑ii′χ4,ii′b8∑M:H†4M,ii′H4M,ii′:+∑a≤c,ii′kmc(ia,i′c)∑JMA†JM(ia,i′c)AJM(ia,i′c).
(1) Equation (1) includes the single-particle Hamiltonian (
Hsp ); theJ=0 andJ=2 pairings (P†0P0 andP†2P2 ); the quadrupole-quadrupole (Q†Q ), octupole-octupole (O†O ), and hexadecapole-hexadecapole (H†H ) terms; and the monopole corrections (Hmc ). In thepn -representation,P†JM,ii′ andA†JM(ia,i′c) are the pair operators, whileQ†2M,ii′ ,O†3M,ii′ , andH†4M,ii′ are the quadrupole, octupole, and hexadecapole operators, respectively, in which i (i′ ) is an index for protons (neutrons). The parametersgJ,ii′ ,χ2,ii′ ,χ3,ii′ ,χ4,ii′ , andkmc(ia,i′c) are the corresponding force strengths, and b is the harmonic-oscillator range parameter.The model space includes five orbits
(0g7/2,1d5/2,1d3/2,2s1/2,0h11/2) for both protons and neutrons. Two extra neutron orbits above theN=82 shell, i.e.,(1f7/2 and2p3/2) , are added to allow neutron cross-shell excitations. We keep the same parameters of single-particle energies and the two-body force strengths used in Ref. [18]. The monopole interactions were found to be crucial for describing nuclear properties, which are entirely responsible for global saturation properties and single-particle behavior. The neutron-rich nuclei near132 Sn can be divided into four quadrants by the crossing ofZ=50 andN=82 . According to nuclei studied previously [15, 22, 30], the monopole corrections are necessary for the hole (particle) nuclei in the southwest (northeast) quadrant of132 Sn.In southwest quadrant (
Z≤50,N≤82 ), the ground state inversion in129 Cd can be well described by the monopole correction between proton orbit0g9/2 and neutron orbit0h11/2 with a strength of –0.40 MeV [17], and it was verified using the recently implemented phase-imaging ion-cyclotron-resonance method [19]. Recently, the ground-state inversions fromN=81 toN=79 were explained for the first time by monopole correction between neutron orbitsh11/2 andd3/2 . Furthermore, this monopole correction has been found in different isotonic chains ofN=79,80,81 , as all being hole nuclei near132 Sn.In the northeast quadrant (
Z≥50,N≥82 ), five monopole terms are used to describe core excitations and high-spin levels [22]. For particle-pole nuclei in the northwest quadrant of132 Sn (Z≥50,N≤82 ), the protons and neutrons occupy the same major shell. The properties of particle-hole nuclei can be well described without additional monopole correlations [28]. Such a situation is confirmed by the electromagnetic transitions of Sb and Te isotopes in the northwest quadrant of132 Sn [28, 31], which is a strict test for shell model calculations. However, in level 17+ of132 Te, the large difference between experiment and theory motivates us to investigate monopole interactions in the particle-hole nuclei region near132 Sn. -
In this part, high-spin levels are investigated with the monopole effects and quadruple-quadruple force in
132−134 Te,131−133 Sb, and130 Sn. The monopole correction alone cannot solve the puzzle of level 17+; it should be combined with quadrupole correction. As shown in Table 1, there are four different types of level 17+ under 10 MeV according to the EPQQM model, i.e., Nos. 1 to 4. The possibility of neutron core excitation (config.1 in Table 1) at 8.189 MeV can be excluded to explain the high-spin level 17+ of132 Te [28]. The given experimental data of134 Te and133 Sb have been reproduced very well as core excitations with a common neutron configuration ofνh−111/2f7/2 (Fig. 1). If we modify the monopole term of Mc(νh−111/2,νf7/2 ) to explain the level 17+ of132 Te, the 17 states of neutron core excitations would catastrophically depart from their corresponding data.Figure 1. (color online) Neutron core excitations in
134 Te and133 Sb nuclei. Corresponding data are from Ref. [32].132 TeEx (MeV) Config. Jπ Th. Exp. No. P(\%) (17 + )8.189 6.166 1. πg27/2νh−311/2f7/2 88 8.779 2. πg7/2h11/2νg−17/2h−111/2 76 9.125 3. πg7/2h11/2νd−15/2h−111/2 72 9.314 4. πh211/2νh−211/2 93 Table 1. 17
+ states in132 Te with main configurations. The data are from Ref. [14].After excluding neutron core excitation, we focus on the level at 8.779 MeV coupled by config.2
πg7/2h11/2νg−17/2h−111/2 . This level can be affected by these monopole terms from this configuration, i.e., Mc(πg7/2 ,πh11/2 ), Mc(νg7/2 ,νh11/2 ), and Mc(πg7/2 ,νh11/2 ). The level at 8.779 MeV is abandoned, since the biggest difference is only 0.227 MeV in level 3− of132 Sb (Fig. 2). For the 17+ level at 9.314 MeV, it has a new monopole term of Mc(πh11/2νh11/2 ). This monopole term has almost no effects in levels 17+ coupled by config.1, 2, and 3.(κ=0.1∼0.9 MeV) , while the level 17+ at 9.314 MeV drops to 8.3 MeV (Fig. 3). The level at 9.134 MeV is excluded too, because all 17+ levels are increased sharply when the strength isκ>0.9 MeV . For the last one coupled by config.3, the suitable monopole termMc(νd5/2,νh11/2) is turned up from this configuration. Its monopole effects are investigated in the states of132 Te from 0+ to 17+ by addingMc(νd5/2,νh11/2) = –2.6 MeV. As shown in Fig. 4, the level 17+ is reduced to 6.311 MeV, but the values of the 5− and 7− levels become far from the corresponding data.Figure 2. (color online) Levels produced by particular configurations in
134 Te,133 Te,132 Sb,131 Sb, and130 Sn nuclei, in comparison with given data [32].Figure 4. (color online) Monopole effects of
Mc(νd5/2,νh11/2) in132 Te. Data marked with stars are from Ref. [32].As shown above, the monopole interaction alone cannot solve the present puzzle, and we focus on the quadrupole-quadrupole force between the proton and neutron (
QQπ,ν ).As shown in Fig. 5, the value of the
QQπ,ν force equals the quadruple-quadruple force strength divided by[1/A(132Sn)]5/3 . The level 17+ coupled by configurationπg7/2h11/2νh−111/2d−15/2 is reduced by 2.121 MeV when the QQ force changes from 300 to –450. The lowest value is 7.107 MeV, which occurs whenQQπν = –350. It seems theQQπ,ν force provides a new method to explain the large difference in level 17+ . As shown in Fig. 6, the datum (17+ ) is reproduced very well withQQπν = –350 andMc(νd5/2,νh11/2)=−0.8 MeV . Furthermore, the values of 5− and 7− levels are close to data (5)− and (7)− . The levels 13− and 14− drop by approximately 1 MeV withQQπ,ν correction(Qc) . The levels 3− and 14− are sensitive to the monopole effects ofMc(νd5/2,νh11/2) . These levels are connected with datum 17+ by theQQπ,ν force and monopole effects ofMc(νd5/2,νh11/2) . If they could be observed experimentally, this would be evidence confirming the datum (17+ ).Figure 5. (color online) Effects of the quadruple-quadruple force between the proton and neutron in level 17
+ of132 Te.For determining the
QQπ,ν correction and monopole termMc(νd5/2,νh11/2) , an alternative of lack data in132 Te is to study theQQπ,ν correction and monopole effects in other nuclei nearby. The monopole effects ofMc(νd5/2,νh11/2) also exist in130 Sn and131 Sb. In130 Sn, the configurationνh−111/2d−15/2 produces levels from 3− to 8− . With theMc(νd5/2,νh11/2) , the configuration percentages have almost no change in levels 3− , 7− , and 8− , while the percentage of level 5− (6− ) drops from 57% (49% ) to 52% (39% ). Level 3− is the first state ofJπ=3− , and its energy decreases by 1.01 MeV when Mc is added. Here, theQQπ,ν force has no effect on one-shell closed nuclei. If this state could be observed near 2.199 MeV, this would be evidence for considering monopole correction in particle-hole nuclei near132 Sn. The same applies to level 8− as the first state ofJπ=8− .In
131 Sb, levels from 1/2− to 21/2− have a main configuration ofπg7/2νh−111/2 . As shown in Fig. 7(b), theQQπ,ν correction has obvious effects on levels 1/2− , 3/2− , 5/2− , 21/2− , and 23/2− , while monopole correction Mc has almost no effects except in the case of level 23/2− . The lowest energy level ofJπ=23/2− drops to 2.570 MeV with Qc and Mc. We are very interested in whether the high-spin state 23/2− can be observed experimentally. This 23/2− level can be used to determine the necessity ofQQπ,ν correction in the particle-hole nuclei region and finally explain the large difference in level (17+ ) of132 Te.Figure 7. (color online)
QQπ,ν force and monopole effects ofMc(νd5/2,νh11/2) in130 Sn and131 Sb. Data marked with stars are from Ref. [32]. -
The monopole effects and high-spin levels in
132−134 Te,131−133 Sb, and130 Sn are investigated. The datum 6.166 MeV of level 17+ in132 Te is excluded from configurationsπg7/2h11/2νg−17/2h−111/2 orπh211/2νh−211/2 . The present work suggests the datum (17+ ) coupled by configurationπg7/2h11/2νd−15/2h−111/2 . Several levels are connected with state (17+ ) by monopole effects ofMc(νd5/2,νh11/2) and the quadruple-quadruple force between the proton and neutron, i.e., 3− (8− ) in130 Sn, 14− in132 Te, and 23/2− in131 Sb. If these states could be observed at lower energies, the lower state (17+ ) of132 Te would be explained with quadruple-quadruple correction between the proton and neutron and the increasing strength of the monopole interaction between neutron orbitsd5/2 andh11/2 .
