# Monopole effects, core excitations, and ${\beta}$ decay in the A = 130 hole nuclei near 132Sn

• The proton and neutron cross-shell excitations across the Z = 50 shell are investigated in the southwest quadrant of 132Sn by large-scale shell-model calculations with extended pairing and multipole-multipole force. The model space allows proton (neutron) core excitations, and both the low- and high-energy states for 130In are well described, as found by comparison with the experimental data. The monopole effects between the proton orbit $g_{9/2}$ and neutron orbit $g_{7/2}$ are studied as the new monopole correction that perfectly reproduces the first 1+ level in 130In. The energy interval of proton (neutron) core excitations in 130In lies in the range of 4.5−6.5 (2.0−4.1) MeV, and the high energy yrast states are predicted as neutron core excitations. The $\beta$ decays are calculated among the A=130 nuclei of 130In, 130Sn and 130Cd.
•  [1] K. L. Kratz, H. Gabelmann, W. Hillebrandt et al, Z. Phys. A, 325: 489 (1986) [2] A. Jungclaus, H. Grawe, S. Nishimura et al, Phys. Rev. C, 94: 024303 (2016) [3] I. Dillmann, K. L. Kratz, A. Wohr et al, Phys. Rev. Lett., 91: 162503 (2003) doi: 10.1103/PhysRevLett.91.162503 [4] G. Martínez-Pinedo and K. Langanke, Phys. Rev. Lett., 83: 4502 (1999) doi: 10.1103/PhysRevLett.83.4502 [5] J. J. Cuenca-García, G. Martínez-Pinedo, K. Langanke et al, Eur. Phys. J. A, 34: 99 (2007) doi: 10.1140/epja/i2007-10477-3 [6] T. Otsuka, T. Suzuki, R. Fujimoto et al, Phys. Rev. Lett., 95: 232502 (2005) doi: 10.1103/PhysRevLett.95.232502 [7] H. K. Wang, Y. Sun, Hua Jin et al, Phys. Rev. C, 88: 054310 (2013) doi: 10.1103/PhysRevC.88.054310 [8] M. Górska, L. Cáceres, H. Grawe et al, Phys. Lett. B, 672: 313 (2009) doi: 10.1016/j.physletb.2009.01.027 [9] F. Minato and C. L. Bai, Phys. Rev. Lett., 110: 122501 (2013) doi: 10.1103/PhysRevLett.110.122501 [10] Y. F. Niu, Z. M. Niu, G. Colò et al, Phys. Rev. Lett., 114: 142501 (2015) doi: 10.1103/PhysRevLett.114.142501 [11] Y. F. Niu, Z. M. Niu, G. Colò et al, Phys. Lett. B, 780: 325 (2018) doi: 10.1016/j.physletb.2018.02.061 [12] P. Bhattacharyya, P. J. Daly, C. T. Zhang et al, Phys. Rev. Lett., 87: 062502 (2001) doi: 10.1103/PhysRevLett.87.062502 [13] K. L. Jones, A. S. Adekola, D. W. Bardayan et al, Nature (London), 465: 454 (2010) doi: 10.1038/nature09048 [14] K. L. Jones, F. M. Nunes, A. S. Adekola et al, Phys. Rev. C, 84: 034601 (2011) doi: 10.1103/PhysRevC.84.034601 [15] H. Watanabe, G. Lorusso, S. Nishimura et al, Phys. Rev. Lett., 111: 152501 (2013) doi: 10.1103/PhysRevLett.111.152501 [16] H. K. Wang, K. Kaneko, and Y. Sun, Phys. Rev. C, 89: 064311 (2014) [17] M. Hasegawa, K. Kaneko, and S. Tazaki, Nucl. Phys. A, 688: 765 (2001) doi: 10.1016/S0375-9474(00)00602-3 [18] K. Kaneko, M. Hasegawa, and T. Mizusaki, Phys. Rev. C, 66: 051306(R) (2002) doi: 10.1103/PhysRevC.66.051306 [19] K. Kaneko, Y. Sun, M. Hasegawa et al, Phys. Rev. C, 78: 064312 (2008) doi: 10.1103/PhysRevC.78.064312 [20] K. Kaneko, Y. Sun, T. Mizusaki et al, Phys. Rev. C, 83: 014320 (2011) doi: 10.1103/PhysRevC.83.014320 [21] Han-Kui Wang, Kazunari Kaneko, Yang Sun et al, Phys. Rev. C, 91: 021303(R) (2015) doi: 10.1103/PhysRevC.91.021303 [22] H. K. Wang, K. Kaneko, Y. Sun et al, Phys. Rev. C, 95: 011304(R) (2017) doi: 10.1103/PhysRevC.95.011304 [23] B. A. Brown and W. D. M. Rae, Nucl. Data Sheets, 120: 115 (2014) doi: 10.1016/j.nds.2014.07.022 [24] H. K. Wang, S. K. Ghorui, K. Kaneko et al, Phys. Rev. C, 96: 054313 (2017) doi: 10.1103/PhysRevC.96.054313 [25] Q. Zhi, E. Caurier, J. J. Cuenca-García et al, Phys. Rev. C, 87: 025803 (2013) [26] Z. M. Niu, Y. F. Niu, H. Z. Liang et al, Phys. Lett. B, 723: 172 (2013) doi: 10.1016/j.physletb.2013.04.048 [27] P. Möller, M. R. Mumpower, T. Kawano et al, Atomic Data and Nuclear Data Tables 125(2019)1-92

Figures(7) / Tables(1)

Get Citation
Han-Kui Wang, Zhi-Hong Li, Cen-Xi Yuan, Zhi-Qiang Chen, Ning Wang, Wei Qin and Yi-Qi He. Monopole effects, core excitations, and ${\beta}$ decay in the A = 130 hole nuclei near 132Sn[J]. Chinese Physics C, 2019, 43(5): 054101. doi: 10.1088/1674-1137/43/5/054101
Han-Kui Wang, Zhi-Hong Li, Cen-Xi Yuan, Zhi-Qiang Chen, Ning Wang, Wei Qin and Yi-Qi He. Monopole effects, core excitations, and ${\beta}$ decay in the A = 130 hole nuclei near 132Sn[J]. Chinese Physics C, 2019, 43(5): 054101.
Milestone
Received: 2018-12-24
Revised: 2019-02-18
Article Metric

Article Views(1204)
PDF Downloads(45)
Cited by(0)
Policy on re-use
To reuse of subscription content published by CPC, the users need to request permission from CPC, unless the content was published under an Open Access license which automatically permits that type of reuse.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

Title:
Email:

## Monopole effects, core excitations, and ${\beta}$ decay in the A = 130 hole nuclei near 132Sn

• 1. College of Physics and Telecommunication Engineering, Zhoukou Normal University, Henan 466000, China
• 2. China Institute of Atomic Energy, P.O. Box 275(10), Beijing 102413, China
• 3. Sino-French Institute of Nuclear Engineering and Technology Sun Yat-Sen University, Zhuhai 519082, China
• 4. School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China
• 5. College of mathematics and statistics, Zhoukou Normal University, Henan 466000, China

Abstract: The proton and neutron cross-shell excitations across the Z = 50 shell are investigated in the southwest quadrant of 132Sn by large-scale shell-model calculations with extended pairing and multipole-multipole force. The model space allows proton (neutron) core excitations, and both the low- and high-energy states for 130In are well described, as found by comparison with the experimental data. The monopole effects between the proton orbit $g_{9/2}$ and neutron orbit $g_{7/2}$ are studied as the new monopole correction that perfectly reproduces the first 1+ level in 130In. The energy interval of proton (neutron) core excitations in 130In lies in the range of 4.5−6.5 (2.0−4.1) MeV, and the high energy yrast states are predicted as neutron core excitations. The $\beta$ decays are calculated among the A=130 nuclei of 130In, 130Sn and 130Cd.

### HTML

#### 1.   Introduction

• 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 r-process waiting-point nuclide 130Cd was first identified by Kratz et al. in 1986 at CERN/ISOLDE [1]. The $\beta$ decay of the semi-magic nucleus 130Cd has been studied at the RIBF facility at the RIKEN Nishina Center, and the energy of the first excited 1+ state of 130In at 2120 keV was confirmed in Ref. [2]. The shell-model 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 132Sn [7]. The core excitation is crucial for studying high energy levels in nuclei close to doubly magic 132Sn. For example, the 17/2+ level is well describedas a core-excited isomer with $T_{1/2}=630(60)ns$ in 131In [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 shell-model interaction and nuclear properties. The shell model calculations were performed to determine the half-lives and neutron-branching probabilities of the r-process waiting-point nuclei at the magic neutron number N = 82, and a good account of all experimentally known half-lives and Q-values is given for these N = 82 r-process waiting-point nuclei [5]. Recently, some frameworks found that the tensor force and particle-vibration coupling play important roles in the beta-decay calculations [9-11], which provide valuable improvements for further shell-model research.

The doubly magic nuclei are very important for the study of nuclear structure. The prompt and delayed $\gamma$-cascades in doubly magic 132Sn and its neighboring 131Sn have been studied at GAMMASPHERE using the 248Cm fission source [12]. By direct observation of single-particle states in odd-mass isotopes close to 132Sn, the doubly magic nature of 132Sn 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 126Pd, 128Pd and 130Cd [2, 15]. Further shell-model calculations are performed on these, and it was concluded that the shell closure persists at the neutron number N = 82 in the neutron-rich region [16]. For the neutron-rich nuclei close to 132Sn, the extended paring-plus-quadrupole model with monopole corrections (EPQQM) [17-20] provides a method to accurately describe both low-lying 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 132Sn to study the monopole effects and core excited states by large-scale shell-model 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 78Ni 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 single-particle energies and the two-body 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 132Sn. The monopole effects and neutron core excitations (NCE) will also be discussed in the present work. The $\beta$ decays among 130Cd, 130In and 130Sn are studied with the quenching factor 0.7. The experimental data are obtained partly from the ENSDF database of NNDC On-line Data Service with cut-off dates of May 11, 2001, May 31, 2008, and May 31, 2008 for 130Sn, 130Cd, and 130In respectively. The shell-model code NUSHELLX@MSU is used for calculations [23].

• #### 2.   The Hamiltonian and modifications

• With the proton-neutron (pn) representation, the EPQQM Hamiltonian [17-20] 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 single-particle Hamiltonian ($H_{\rm sp}$), the J = 0 and J = 2 pairing ($P_{0}^{\dagger}P_{0}$ and $P_{2}^{\dagger}P_{2}$), the quadrupole-quadrupole ($Q^{\dagger}Q$), the octupole-octupole ($O^{\dagger}O$), the 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, 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 harmonic-oscillator range parameter. The two-body force strengths that suit the present particle-hole model space are listed in Table 1.

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}, $$\nu h_{11/2})= -0.4 MeV are employed to modify the N = 82 shell gap. The M_c2 is also necessary for obtaining the right ground state of 129Cd [22]. The M_c3\!\equiv\! k_{\rm mc}(\pi g_{9/2}, \nu g_{7/2})\!=\! −1.0 MeV is the new monopole correction introduced in the present work. As shown in Fig. 1, adding the monopole corrections of M_c1 and M_c2 has no influence on low-lying levels of 130Sn and 130Cd, while the M_c2 shifts up the 3+, 5+ and 1+ levels in 130In. The fact that the datum 1+ level at 2.120 MeV is much lower than the theoretical value in the present work is the experimental base for adding M_c3 . The orbit \pi g_{9/2} ( \nu g_{7/2} ) is full of particles in the configuration of low-lying levels at 130Sn (130Cd), however the M_c3\equiv$$ 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 130In. Hence, the Hamiltonian including $M_c3$ has no effects on low-lying levels of 130Sn and 130Cd, however it significantly shifts down the 1+ level in 130In. 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 core-excitations [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 low-lying levels of hole nuclei 130Sn, 130Cd and 130In 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 128Pd [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 low-lying levels of 130Sn, it breaks the degeneracy in 4+, 6+, and 8+ levels of 130Cd, and also shifts down the 5+ level in 130In to close to the corresponding datum in the experiment. In the research of the A = 129 hole nuclei near 132Sn, two zero-value 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. $$\left.|V| p_{1/2}, g_{9/2}, \right> _{J=5}^{\pi}= -0.22 . The M_e2 has no effects on the low-lying levels of 130Sn, and little effect on 130Cd. It has obvious effects on shifting down the 5+ level in 130In. During the study of Sb isotopes in the particle-hole nuclei near 132Sn, the new modification is needed in the three interaction matrix elements (marked as M_e3 ) by the amount of \Delta \langle h_{11/2}, h_{11/2}|V|h_{11/2}, h_{11/2}\rangle^{\nu}=$$-0.15, -0.15, +0.2$ (MeV) for J = 6, 8, 10, respectively. Although the $M_e3$ is used in the northeastern quadrant of 132Sn [24], it also works well in the present hole-nuclei region. As shown in Fig. 1(a), the $M_e3$ not only breaks the degeneracy in 6+, 8+ and 10+ levels in 130Sn, but also gives the consistent order with experimental data.

Reference (27)

### 目录

/

DownLoad:  Full-Size Img  PowerPoint