
The study of nuclear structure of neutron rich nuclei near the A = 130 mass region is fundamentally important for both nuclear physics and astrophysics. The classical N = 82 rprocess waitingpoint nuclide ^{130}Cd was first identified by Kratz et al. in 1986 at CERN/ISOLDE [1]. The
$ \beta $ decay of the semimagic nucleus ^{130}Cd has been studied at the RIBF facility at the RIKEN Nishina Center, and the energy of the first excited 1^{+} state of ^{130}In at 2120 keV was confirmed in Ref. [2]. The shellmodel calculations produce the 1^{+} level by undervaluation of 550–750 keV [3, 4]. The obvious difference between shell model theory and experiment was improved by introducing monopole corrections to the employed Hamiltonian [5].The monopole interaction plays not only a significant role in the shell evolution owing to the monopole shift when valence nucleons occupy certain orbits [6], but also modifies the high core excitations in hole nuclei close to doubly magic ^{132}Sn [7]. The core excitation is crucial for studying high energy levels in nuclei close to doubly magic ^{132}Sn. For example, the 17/2^{+} level is well describedas a coreexcited isomer with
$ T_{1/2}=630(60)ns $ in ^{131}In [8]. Besides the core excitations, the$ \beta $ decay also plays an important role in nuclear physics, astrophysics and particle physics, and provides crucial information about the shellmodel interaction and nuclear properties. The shell model calculations were performed to determine the halflives and neutronbranching probabilities of the rprocess waitingpoint nuclei at the magic neutron number N = 82, and a good account of all experimentally known halflives and Qvalues is given for these N = 82 rprocess waitingpoint nuclei [5]. Recently, some frameworks found that the tensor force and particlevibration coupling play important roles in the betadecay calculations [911], which provide valuable improvements for further shellmodel research.The doubly magic nuclei are very important for the study of nuclear structure. The prompt and delayed
$ \gamma $ cascades in doubly magic ^{132}Sn and its neighboring ^{131}Sn have been studied at GAMMASPHERE using the ^{248}Cm fission source [12]. By direct observation of singleparticle states in oddmass isotopes close to ^{132}Sn, the doubly magic nature of ^{132}Sn has been reconfirmed [13, 14]. As a strong signal of shell closure, the validity of the seniority has been predicted in earlier theoretical calculations and confirmed through experimental observation of 8^{+} seniority isomeric states in ^{126}Pd, ^{128}Pd and ^{130}Cd [2, 15]. Further shellmodel calculations are performed on these, and it was concluded that the shell closure persists at the neutron number N = 82 in the neutronrich region [16]. For the neutronrich nuclei close to ^{132}Sn, the extended paringplusquadrupole model with monopole corrections (EPQQM) [1720] provides a method to accurately describe both lowlying states and core excitations in a consistent manner [7, 16, 21].In the present work, the EPQQM are applied to the hole nuclei at the southwestern quadrant of ^{132}Sn to study the monopole effects and core excited states by largescale shellmodel calculations. The model space consists of five neutron orbits (
$ 0g_{7/2}, 1d_{5/2}, 2s_{1/2}, 0h_{11/2}, 1d_{3/2} $ ) and four proton orbits ($ 0f_{5/2}, 1p_{3/2}, 1p_{1/2}, 0g_{9/2} $ ) with ^{78}Ni as the closed core. In addition, two neutron orbits ($ 1f_{7/2} $ ,$ 2p_{3/2} $ ) above the N = 82 shell gap and two proton orbits ($ 0g_{7/2} $ ,$ 1d_{5/2} $ ) above the Z = 50 shell gap are included for allowing both proton and neutron core excitations. The proton (neutron) core excitations were restricted, such that only one proton (neutron) was allowed to excite across the Z = 50 (N = 82) shell gap. The singleparticle energies and the twobody force strengths employed in the present work are consistent with our previous paper [21, 22]. The proton core excitations (PCE) are firstly discussed in the hole nuclei close to ^{132}Sn. The monopole effects and neutron core excitations (NCE) will also be discussed in the present work. The$ \beta $ decays among ^{130}Cd, ^{130}In and ^{130}Sn are studied with the quenching factor 0.7. The experimental data are obtained partly from the ENSDF database of NNDC Online Data Service with cutoff dates of May 11, 2001, May 31, 2008, and May 31, 2008 for ^{130}Sn, ^{130}Cd, and ^{130}In respectively. The shellmodel code NUSHELLX@MSU is used for calculations [23]. 
With the protonneutron (pn) representation, the EPQQM Hamiltonian [1720] is given as follows:
$ \begin{split} H =& H_{\rm sp} + H_{P_0} + H_{P_2} + H_{QQ} + H_{OO} + H_{HH} + H_{\rm mc} \\ = & \displaystyle\sum_{\alpha, i} \varepsilon_a^i c_{\alpha, i}^\dagger c_{\alpha, i}  \displaystyle\frac{1}{2} \displaystyle\sum_{J=0, 2} \displaystyle\sum_{ii'} g_{J, ii'} \displaystyle\sum_{M} P^\dagger_{JM, ii'} P_{JM, ii'} \\ &  \frac{1}{2} \displaystyle\sum_{ii'} \displaystyle\frac{\chi_{2, ii'}}{b^4} \displaystyle\sum_M :Q^\dagger_{2M, ii'} Q_{2M, ii'}: \\ &  \displaystyle\frac{1}{2} \displaystyle\sum_{ii'} \displaystyle\frac{\chi_{3, ii'}}{b^6} \displaystyle\sum_M :O^\dagger_{3M, ii'} O_{3M, ii'}: \\ &  \displaystyle\frac{1}{2} \displaystyle\sum_{ii'} \displaystyle\frac{\chi_{4, ii'}}{b^8} \displaystyle\sum_M :H^\dagger_{4M, ii'} H_{4M, ii'}: \\ & + \displaystyle\sum_{a \leqslant c, ii'} k_{\rm mc}(ia, i'c) \displaystyle\sum_{JM}A^\dagger_{JM}(ia, i'c) A_{JM}(ia, i'c). \end{split} $
(1) Equation (1) includes the singleparticle Hamiltonian (
$ H_{\rm sp} $ ), the J = 0 and J = 2 pairing ($ P_{0}^{\dagger}P_{0} $ and$ P_{2}^{\dagger}P_{2} $ ), the quadrupolequadrupole ($ Q^{\dagger}Q $ ), the octupoleoctupole ($ O^{\dagger}O $ ), the hexadecapolehexadecapole ($ H^{\dagger}H $ ) terms, and the monopole corrections ($ H_{\rm mc} $ ). In the pnrepresentation,$ 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, where i (i') depict the indices 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 harmonicoscillator range parameter. The twobody force strengths that suit the present particlehole model space are listed in Table 1.ii' $ g_{0,ii'} $ $ g_{2,ii'} $ $ \chi_{2,ii'} $ $ \chi_{3,ii'} $ $ \chi_{4,ii'} $ pp 0.136 0.038 0.102 0.032 0.0015 nn 0.117 0.035 0.140 0.004 0.0008 pn 0 0 0.082 0 0.0009 Table 1. Twobody force strengths (in MeV) used in the present calculation.
In our previous papers, the monopole corrections of
$ M_c1\equiv k_{\rm mc}(\nu h_{11/2}, \nu f_{7/2})=0.52 $ MeV and$M_c2\equiv k_{\rm mc}(\pi g_{9/2}, $ $ \nu h_{11/2})= 0.4 $ MeV are employed to modify the N = 82 shell gap. The$ M_c2 $ is also necessary for obtaining the right ground state of ^{129}Cd [22]. The$ M_c3\!\equiv\! k_{\rm mc}(\pi g_{9/2}, \nu g_{7/2})\!=\! $ −1.0 MeV is the new monopole correction introduced in the present work. As shown in Fig. 1, adding the monopole corrections of$ M_c1 $ and$ M_c2 $ has no influence on lowlying levels of ^{130}Sn and ^{130}Cd, while the$ M_c2 $ shifts up the 3^{+}, 5^{+} and 1^{+} levels in ^{130}In. The fact that the datum 1^{+} level at 2.120 MeV is much lower than the theoretical value in the present work is the experimental base for adding$ M_c3 $ . The orbit$ \pi g_{9/2} $ ($ \nu g_{7/2} $ ) is full of particles in the configuration of lowlying levels at ^{130}Sn (^{130}Cd), however the$ M_c3\equiv$ $ k_{\rm mc}(\pi g_{9/2}, \nu g_{7/2}) $ has effects only when the orbits$ \pi g_{9/2} $ and$ \nu g_{7/2} $ are both lacking particles, as in the 1^{+} level with a configuration of$ \pi g^{1}_{9/2} \nu g^{1}_{7/2} $ in ^{130}In. Hence, the Hamiltonian including$ M_c3 $ has no effects on lowlying levels of ^{130}Sn and ^{130}Cd, however it significantly shifts down the 1^{+} level in ^{130}In. This represents a neat reduction of the theoretical value and the corresponding datum. Unlike the addition of$ M_c1 $ and$ M_c2 $ to modify high coreexcitations [7], the purpose of adding$ M_c3 $ is not for core excitation, as no effect was found on core excitations mentioned in the present work by adding$ M_c3 $ .Figure 1. (color online) Effects of monopole corrections (
$ M_c $ ), matrix element ($ M_e $ ) modifications and allowing the neutron (proton) core excitations (NCE & PCE) in the lowlying levels of hole nuclei ^{130}Sn, ^{130}Cd and ^{130}In in comparison with corresponding data. The$ M_c $ and$ M_e $ are added to the initial Hamiltonian (00) one by one, and their definition and value of the$ M_c $ and$ M_e $ are given in the text.The matrix element of J = 4 in the proton orbit
$ g_{9/2} $ is increased by –0.18 MeV to obtain the right order of 6^{+} and 8^{+} levels in ^{128}Pd [16]. For simplified notation, this modification is marked as$ M_e1 $ . As shown in Fig. 1(a), although the$ M_e1 $ has no effects on the lowlying levels of ^{130}Sn, it breaks the degeneracy in 4^{+}, 6^{+}, and 8^{+} levels of ^{130}Cd, and also shifts down the 5^{+} level in ^{130}In to close to the corresponding datum in the experiment. In the research of the A = 129 hole nuclei near ^{132}Sn, two zerovalue matrix elements were given new values in the Hamiltonian (marked as$ M_e2 $ ),$ \left< p_{1/2}, g_{9/2}, V p_{1/2}, g_{9/2}, \right>_{J=4}^{\pi} = $ 0.32 and$\left< p_{1/2}, g_{9/2},\right. $ $ \left.V p_{1/2}, g_{9/2}, \right> _{J=5}^{\pi}= 0.22 $ . The$ M_e2 $ has no effects on the lowlying levels of ^{130}Sn, and little effect on ^{130}Cd. It has obvious effects on shifting down the 5^{+} level in ^{130}In. During the study of Sb isotopes in the particlehole nuclei near ^{132}Sn, the new modification is needed in the three interaction matrix elements (marked as$ M_e3 $ ) by the amount of$ \Delta \langle h_{11/2}, h_{11/2}Vh_{11/2}, h_{11/2}\rangle^{\nu}= $ $0.15, 0.15, +0.2 $ (MeV) for J = 6, 8, 10, respectively. Although the$ M_e3 $ is used in the northeastern quadrant of ^{132}Sn [24], it also works well in the present holenuclei region. As shown in Fig. 1(a), the$ M_e3 $ not only breaks the degeneracy in 6^{+}, 8^{+} and 10^{+} levels in ^{130}Sn, but also gives the consistent order with experimental data.
Monopole effects, core excitations, and ${\beta}$ decay in the A = 130 hole nuclei near ^{132}Sn
 Received Date: 20181224
 Accepted Date: 20190218
 Available Online: 20190501
Abstract: The proton and neutron crossshell excitations across the Z = 50 shell are investigated in the southwest quadrant of ^{132}Sn by largescale shellmodel calculations with extended pairing and multipolemultipole force. The model space allows proton (neutron) core excitations, and both the low and highenergy states for ^{130}In are well described, as found by comparison with the experimental data. The monopole effects between the proton orbit