Monopole effects and high-spin levels in neutron-rich ¹³²Te

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 at $ A\sim130 $, which is formed through the rapid neutron capture process [1, 2]. The properties of doubly magic $ ^{132} $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 of $ ^{134} $Te in the 4$ ^+ $ state was measured, which provides direct insight into the single particle structure [9]. The state (17$ ^+ $) of $ ^{132} $Te was observed to be 6.166 MeV using the reaction $ ^{9} $Be($ ^{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 in $ ^{128} $Cd is predicted as a spin-trap isomer feeding the known 16$ ^+ $ of $ ^{128} $In through $ \beta^- $ 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 in $ ^{133} $Ba requires the cross-shell orbits lying above the energy gap $ N = 82 $. As reported in Ref. [21], the interactions without core excitations cannot provide the B(E1) value for the transition from $ J^{\pi} = 19/2^+ $ to $ J^{\pi} = 19/2^- $.

This model has an advantage for studying monopole effects by employing monopole correction (Mc) terms. For example, in the southwest quadrant ($ Z \leq 50, N \leq 82 $) of $ ^{132} $Sn, the level spectra and the energy gap across $ N = 82 $ can be modified by monopole correction between neutron orbit $ h_{11/2} $ and $ f_{7/2} $ [15]. In the northeast quadrant ($ Z \geq 50, N \geq 82 $), five monopole terms are used to describe core excitations and high-spin levels, and the states 2$ ^- $ and 9$ ^- $ in $ ^{136,138} $Te are predicted as a spin-trap structure coupled by the neutron intruder orbit $ i_{13/2} $ [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 of $ ^{132} $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]:

Equation (1) includes the single-particle Hamiltonian ($ H_{\rm sp} $); the $ J=0 $ and $ J=2 $ pairings ($ P_{0}^{\dagger}P_{0} $ and $ P_{2}^{\dagger}P_{2} $); the quadrupole-quadrupole ($ Q^{\dagger}Q $), octupole-octupole ($ O^{\dagger}O $), and hexadecapole-hexadecapole ($ H^{\dagger}H $) terms; and the monopole corrections ($ H_{\rm mc} $). In the $ pn $-representation, $ P^\dagger_{JM,ii'} $ and $ A^\dagger_{JM}(ia,i'c) $ are the pair operators, while $ Q^\dagger_{2M,ii'} $, $ O^\dagger_{3M,ii'} $, and $ H^\dagger_{4M,ii'} $ are the quadrupole, octupole, and hexadecapole operators, respectively, in which i ($ i' $) is an index for protons (neutrons). The parameters $ g_{J,ii'} $, $ \chi_{2,ii'} $, $ \chi_{3,ii'} $, $ \chi_{4,ii'} $, and $ k_{\rm mc}(ia,i'c) $ are the corresponding force strengths, and b is the harmonic-oscillator range parameter.

The model space includes five orbits $ (0g_{7/2}, 1d_{5/2}, 1d_{3/2}, 2s_{1/2}, 0h_{11/2}) $ for both protons and neutrons. Two extra neutron orbits above the $ N=82 $ shell, i.e., $ (1f_{7/2} $ and $ 2p_{3/2}) $, 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 near $ ^{132} $Sn can be divided into four quadrants by the crossing of $ Z = 50 $ and $ N = 82 $. According to nuclei studied previously [15, 22, 30], the monopole corrections are necessary for the hole (particle) nuclei in the southwest (northeast) quadrant of $ ^{132} $Sn.

In southwest quadrant ($ Z \leq 50, N \leq 82 $), the ground state inversion in $ ^{129} $Cd can be well described by the monopole correction between proton orbit $ 0g_{9/2} $ and neutron orbit $ 0h_{11/2} $ 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 from $ N = 81 $ to $ N = 79 $ were explained for the first time by monopole correction between neutron orbits $ h_{11/2} $ and $ d_{3/2} $. Furthermore, this monopole correction has been found in different isotonic chains of $ N = 79,80,81 $, as all being hole nuclei near $ ^{132} $Sn.

In the northeast quadrant ($ Z \geq 50, N \geq 82 $), five monopole terms are used to describe core excitations and high-spin levels [22]. For particle-pole nuclei in the northwest quadrant of $ ^{132} $Sn ($ Z \geq 50, N \leq 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 of $ ^{132} $Sn [28, 31], which is a strict test for shell model calculations. However, in level 17$ ^+ $ of $ ^{132} $Te, the large difference between experiment and theory motivates us to investigate monopole interactions in the particle-hole nuclei region near $ ^{132} $Sn.

In this part, high-spin levels are investigated with the monopole effects and quadruple-quadruple force in $ ^{132-134} $Te, $ ^{131-133} $Sb, and $ ^{130} $Sn. The monopole correction alone cannot solve the puzzle of level 17⁺; it should be combined with quadrupole correction. As shown in Table 1, there are four different types of level 17$ ^+ $ 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$ ^+ $ of $ ^{132} $Te [28]. The given experimental data of $ ^{134} $Te and $ ^{133} $Sb have been reproduced very well as core excitations with a common neutron configuration of $ \nu h^{-1}_{11/2} f_{7/2} $ (Fig. 1). If we modify the monopole term of Mc($ \nu h^{-1}_{11/2}, \nu f_{7/2} $) to explain the level 17$ ^+ $ of $ ^{132} $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 and $^{133}$Sb nuclei. Corresponding data are from Ref. [32].

$^{132}$Te	$E_x$	(MeV)	Config.
$J^{\pi}$	Th.	Exp.	No.	P(\%)
(17$^+$)	8.189	6.166	1. $\pi g^2_{7/2}\nu h^{-3}_{11/2} f_{7/2}$	88
	8.779		2. $\pi g_{7/2} h_{11/2} \nu g^{-1}_{7/2} h^{-1}_{11/2}$	76
	9.125		3. $\pi g_{7/2} h_{11/2} \nu d^{-1}_{5/2} h^{-1}_{11/2}$	72
	9.314		4. $\pi h^2_{11/2} \nu h^{-2}_{11/2}$	93

Table 1.  17$^+$ states in $^{132}$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 $ \pi g_{7/2} h_{11/2} \nu g^{-1}_{7/2} h^{-1}_{11/2} $. This level can be affected by these monopole terms from this configuration, i.e., Mc($ \pi g_{7/2} $,$ \pi h_{11/2} $), Mc($ \nu g_{7/2} $,$ \nu h_{11/2} $), and Mc($ \pi g_{7/2} $,$ \nu h_{11/2} $). The level at 8.779 MeV is abandoned, since the biggest difference is only 0.227 MeV in level 3$ ^- $ of $ ^{132} $Sb (Fig. 2). For the 17$ ^+ $ level at 9.314 MeV, it has a new monopole term of Mc($ \pi h_{11/2} \nu h_{11/2} $). This monopole term has almost no effects in levels 17$ ^+ $ coupled by config.1, 2, and 3. $(\kappa = 0.1 \sim 0.9~\rm MeV)$, 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 is $\kappa > 0.9~\rm MeV$. For the last one coupled by config.3, the suitable monopole term $ Mc(\nu d_{5/2},\nu h_{11/2}) $ is turned up from this configuration. Its monopole effects are investigated in the states of $ ^{132} $Te from 0$ ^+ $ to 17$ ^+ $ by adding $ Mc(\nu d_{5/2},\nu h_{11/2}) $ = –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, and $^{130}$Sn nuclei, in comparison with given data [32].

Figure 3.  (color online) Monopole effects of Mc($\pi h_{11/2} \nu h_{11/2}$) in level $17^+$ of $^{132}$Te.

Figure 4.  (color online) Monopole effects of $Mc(\nu d_{5/2},\nu h_{11/2})$ in $^{132}$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 ($ QQ_{\pi,\nu} $).

As shown in Fig. 5, the value of the$ QQ_{\pi,\nu} $ force equals the quadruple-quadruple force strength divided by $ [1/A(^{132}Sn)]^{5/3} $. The level 17$ ^+ $ coupled by configuration $ \pi g_{7/2}h_{11/2} \nu h^{-1}_{11/2} d^{-1}_{5/2} $ is reduced by 2.121 MeV when the QQ force changes from 300 to –450. The lowest value is 7.107 MeV, which occurs when $ QQ_{\pi \nu} $ = –350. It seems the $ QQ_{\pi,\nu} $ 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 with $ QQ_{\pi \nu} $ = –350 and $Mc(\nu d_{5/2}, \nu h_{11/2}) = -0.8~\rm MeV$. 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 with $ QQ_{\pi,\nu} $ correction $(Q_c) $. The levels 3$ ^- $ and 14$ ^- $ are sensitive to the monopole effects of $ Mc(\nu d_{5/2},\nu h_{11/2}) $. These levels are connected with datum 17$ ^+ $ by the $ QQ_{\pi,\nu} $ force and monopole effects of $ Mc(\nu d_{5/2},\nu h_{11/2}) $. If they could be observed experimentally, this would be evidence confirming the datum (17$ ^+ $).

Figure 5.  (color online) Effects of the quadruple-quadruple force between the proton and neutron in level 17$^+$ of $^{132}$Te.

Figure 6.  (color online) $QQ_{\pi,\nu}$ force and monopole effects of $Mc(\nu d_{5/2},\nu h_{11/2})$ in $^{132}$Te. Data marked with stars are from Refs. [14, 32].

For determining the $ QQ_{\pi,\nu} $ correction and monopole term $ Mc(\nu d_{5/2},\nu h_{11/2}) $, an alternative of lack data in $ ^{132} $Te is to study the $ QQ_{\pi,\nu} $ correction and monopole effects in other nuclei nearby. The monopole effects of $ Mc(\nu d_{5/2},\nu h_{11/2}) $ also exist in $ ^{130} $Sn and $ ^{131} $Sb. In $ ^{130} $Sn, the configuration $ \nu h^{-1}_{11/2} d^{-1}_{5/2} $ produces levels from 3$ ^- $ to 8$ ^- $. With the $ Mc(\nu d_{5/2},\nu h_{11/2}) $, the configuration percentages have almost no change in levels 3$ ^- $, 7$ ^- $, and 8$ ^- $, while the percentage of level 5$ ^- $ (6$ ^- $) drops from 57$ \% $ (49$ \% $) to 52$ \% $ (39$ \% $). Level 3$ ^- $ is the first state of $ J^{\pi} = 3^- $, and its energy decreases by 1.01 MeV when Mc is added. Here, the $ QQ_{\pi,\nu} $ 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 near $ ^{132} $Sn. The same applies to level 8$ ^- $ as the first state of $ J^{\pi} = 8^- $.

In $ ^{131} $Sb, levels from 1/2$ ^- $ to 21/2$ ^- $ have a main configuration of $ \pi g_{7/2} \nu h^{-1}_{11/2} $. As shown in Fig. 7(b), the $ QQ_{\pi,\nu} $ 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 of $ J^{\pi} = 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 of $ QQ_{\pi,\nu} $ correction in the particle-hole nuclei region and finally explain the large difference in level (17$ ^+ $) of $ ^{132} $Te.

Figure 7.  (color online) $QQ_{\pi,\nu}$ force and monopole effects of $Mc(\nu d_{5/2},\nu h_{11/2})$ in $^{130}$Sn and $^{131}$Sb. Data marked with stars are from Ref. [32].

The monopole effects and high-spin levels in $ ^{132-134} $Te, $ ^{131-133} $Sb, and $ ^{130} $Sn are investigated. The datum 6.166 MeV of level 17$ ^+ $ in $ ^{132} $Te is excluded from configurations $ \pi g_{7/2} h_{11/2} \nu g^{-1}_{7/2} h^{-1}_{11/2} $ or $ \pi h^2_{11/2} \nu h^{-2}_{11/2} $. The present work suggests the datum (17$ ^+ $) coupled by configuration $ \pi g_{7/2} h_{11/2} \nu d^{-1}_{5/2} h^{-1}_{11/2} $. Several levels are connected with state (17$ ^+ $) by monopole effects of $ Mc(\nu d_{5/2},\nu h_{11/2}) $ and the quadruple-quadruple force between the proton and neutron, i.e., 3$ ^- $ (8$ ^- $) in $ ^{130} $Sn, 14$ ^- $ in $ ^{132} $Te, and 23/2$ ^- $ in $ ^{131} $Sb. If these states could be observed at lower energies, the lower state (17$ ^+ $) of $ ^{132} $Te would be explained with quadruple-quadruple correction between the proton and neutron and the increasing strength of the monopole interaction between neutron orbits $ d_{5/2} $ and $ h_{11/2} $.