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Shape coexistence in 76Se within the neutron–proton interacting boson model

  • Cheng-Fu Mu(穆成富) , ∗∗ ,
  • Da-Li Zhang(张大立) , ∗∗
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  • Department of physics, Huzhou University, Huzhou 313000, Zhejiang, China

∗∗ Authors to whom any correspondence should be addressed.

Received date: 2021-10-12

  Revised date: 2021-11-09

  Accepted date: 2021-12-03

  Online published: 2022-03-10

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© 2022 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

We have investigated the low-lying energy spectrum and electromagnetic transition strengths in even–even 76Se using the proton–neutron interacting boson model (IBM-2). The theoretical calculation for the energy levels and E2 and M1 transition strengths is in good agreement with the experimental data. Specifically, the excitation energy and E2 transition of ${0}_{2}^{+}$ state, which is intimately associated with shape coexistence, can be accurately reproduced. The analysis on low-lying states and the key structure indicators R1, R2, R3 and R4 and M1 transitions indicates that there is a coexistence between spherical shape and γ-soft shape in 76Se.

Cite this article

Cheng-Fu Mu(穆成富) , Da-Li Zhang(张大立) . Shape coexistence in 76Se within the neutron–proton interacting boson model[J]. Communications in Theoretical Physics, 2022 , 74(2) : 025302 . DOI: 10.1088/1572-9494/ac3fb0

1. Introduction

In recent years, the shapes and shape evolution of selenium isotopes have been discussed in many works [16], in which the shapes of nuclei rapidly evolves from a prolate deformation around the line of β stability towards oblate deformation near the line of N = Z [7]. The experimental results in [8] extended the yrast band of 76Se from spin 12+ at 5.429 MeV to 22+ at 13.679 MeV and γ band from 10+ at 4.685 MeV to 19+ at 11.145 MeV, offering a chance for observation of the second band crossing. The theoretical and experimental results in [911] found that the triaxial degree of freedom plays an important role in 76Se. Henderson et al extracted some electric quadrupole matrix elements and quadrupole moment of ${2}_{1}^{+}$ state which is consistent with a prolate deformation, but the ground state exhibits significant triaxiality [11]. The dipole response of 76Se in the energy interval from 4 to 9MeV has been discussed in terms of a polarized photon scattering technique [12]. H T Fortune investigated the coexistence and mixing using a simple two-state mixing model in 76Se [13]. It is long believed that 76Se nucleus can provide a platform to examine the shape coexistence. Shape phase transition and shape coexistence are a longstanding hot topic in nuclear physics [1427]. In the [28], Mukhopadhyay et al observed the band structure based on the ${0}_{2}^{+}$ state at 1.122 MeV, the members of this band include the ${2}_{3}^{+}$ state at 1.788 MeV and ${4}_{4}^{+}$ state at 2.618 MeV, the experimental results support the shape coexistence in 76Se.
Theoretical studies of selenium isotopes have been carried for a long time. U Kaup et al analyzed the band structure of selenium isotopes using the interacting boson model (IBM) [29]. Budaca et al in [30] discussed the shape coexistence in Se isotopes in terms of a separable Bohr model using a double-well potential. K Nomura et al studied the structural evolution in selenium nuclei by means of the mapped IBM based on the Gogny energy density functional [2]. In this paper, we extend our previous studies of IBM-2 on the shape coexistence to 76Se nucleus [3141], and extract the associated signatures of shape coexistence by analyzing the low-lying level scheme and M1 transition transition rates in addition to E2 transitons. We expect to improve the description of ${0}_{2}^{+}$ state in the framework of IBM-2 which is important for the shape coexistence.
The outline of this paper is listed as follows. In section 2, the IBM-2 Hamiltonian, as well as E2 and M1 operators used in this paper are briefly introduced. In section 3, we determine the model parameters, and compare our calculated results including E2 and M1 transitions with the experimental ones. The conclusion is presented in section 4.

2. Theoretical framework

In the original interaction boson model (IBM-1) proposed by Arima and Iachello, the collective properties of an even–even nucleus can be described by the coherent states of s bosons (angular momentum l = 0 )and d bosons (angular momentum l = 2 ). Therefore, the IBM can be considered as one kind of truncation of shell model. The IBM-1 has U(6) symmetry group [42]. By differentiating neutron bosons from proton bosons, the IBM-1 is extended to proton–neutron interacting boson model or IBM-2 [4347]. The IBM-2 possesses a much richer phase structure and demonstrates a more predictive power in dealing with nuclear experimental data. In the IBM-2, the triaxial nuclear shape naturally occurs when the axes of a proton fluid with prolate deformation and a neutron fluid with oblate deformation are orthogonal to each other [48]. Especially, the IBM-2 predicts a new class of collective state with mixed symmetry in protons and neutrons called mixed-symmetry (MS) states, for the detailed discussion refer to [49, 50]. The other states are fully symmetric, generally known as FS states, which also can be equivalently described by IBM-1 [51]. The IBM-2 Hamiltonian can include many interaction terms. In this paper, we adopt the following terms of Hamiltonian which gives the major contribution to 76Se, more details are referred to in the following works [39, 4850, 52],
$\begin{eqnarray}\begin{array}{l}\hat{H}={\varepsilon }_{\pi }{\hat{n}}_{d\pi }+{\varepsilon }_{\nu }{\hat{n}}_{d\nu }+{\kappa }_{\pi \nu }{\hat{Q}}_{\pi }\cdot {\hat{Q}}_{\nu }\\ +{\omega }_{\pi \pi }{\hat{L}}_{\pi }\cdot {\hat{L}}_{\pi }+{\hat{M}}_{\pi \nu }\\ +\displaystyle \frac{1}{2}\sum _{\rho =\pi ,\nu }\sum _{L=0,2,4}{(2L+1)}^{\tfrac{1}{2}}{c}_{\rho }^{(L)}\\ ({[{d}_{\rho }^{\dagger }{d}_{\rho }^{\dagger }]}^{(L)}\cdot {[{\tilde{d}}_{\rho }{\tilde{d}}_{\rho }]}^{(L)}{)}_{0}^{(0)},\end{array}\end{eqnarray}$
where the first two terms ${\varepsilon }_{d\rho }{\hat{n}}_{d\rho }$ denote the energy terms of single d-boson with ρ = π for proton bosons and ρ = ν for neutron bosons. ${\hat{n}}_{d\rho }={d}_{\rho }^{\dagger }\cdot \tilde{{d}_{\rho }}\ $ is the number operator of d-boson. The third term stands for the quadrupole–quadrupole interaction between proton bosons and neutron bosons. The fourth term is the dipole interaction between the proton bosons, here we do not include the dipole interaction between neutron bosons in order to reduce free parameters. The explicit form of quadrupole operator ${\hat{Q}}_{\rho }$ is ${\hat{Q}}_{\rho }={({s}_{\rho }^{\dagger }{\tilde{d}}_{\rho }+{d}_{\rho }^{\dagger }{s}_{\rho })}^{(2)}$ $+{\chi }_{\rho }{({d}_{\rho }^{\dagger }{\tilde{d}}_{\rho })}^{(2)}\,$ with κπν being the strength parameter. The parameter χρ is related with the type of the deformation (χρ < 0 forprolate shape, χρ > 0 for oblate shape and χρ ≈ 0 for γ-unstable shape). The angular momentum operator Lρ has its explicit form $\hat{{L}_{\rho }}=\sqrt{10}[{d}_{\rho }^{\dagger }$ $\cdot \tilde{{d}_{\rho }}{]}^{(1)}$. The Majorana operator has the form ${\hat{M}}_{\pi \nu }={\lambda }_{2}({s}_{\pi }^{\dagger }{d}_{\nu }^{\dagger }-{s}_{\nu }^{\dagger }$ ${d}_{\pi }^{\dagger }{)}^{(2)}$ $\cdot {({s}_{\pi }{\tilde{d}}_{\nu }-{s}_{\nu }{\tilde{d}}_{\pi })}^{(2)}\,$ $+{\sum }_{k=\mathrm{1,3}}{\lambda }_{k}({d}_{\pi }^{\dagger }$ ${d}_{\nu }^{\dagger }{)}^{(k)}$ $\cdot {({\tilde{d}}_{\pi }{\tilde{d}}_{\nu })}^{(k)}$. with three Majorana parameters λk (k=1, 2, 3) describing the strength of the Majorana interaction. The last term corresponds to the interaction between like-bosons.
In the IBM-2 model, the E2 and M1 transition operators are expressed as:
$\begin{eqnarray}{\hat{T}}^{(E2)}={e}_{\pi }{\hat{Q}}_{\pi }+{e}_{\nu }{\hat{Q}}_{\nu },\ \end{eqnarray}$
$\begin{eqnarray}{\hat{T}}^{(M1)}=\sqrt{3/4\pi }({g}_{\nu }{\hat{L}}_{\nu }+{g}_{\pi }{\hat{L}}_{\pi }).\ \end{eqnarray}$
where the effective quadrupole charge eρ and the gyromagnetic ratio gρ are independent parameters and determined by fitting the experimental data.
Once one obtains the boson wave functions of a given nucleus, the reduced E2 and M1 electromagnetic transition probabilities can be easily computed through the quantum mechanics principle. The E2 transition strength B(E2) and M1 transition strength B(M1) are obtained by [38, 48, 52]
$\begin{eqnarray}B(E2,J\to J^{\prime} )=\displaystyle \frac{1}{2J+1}| \langle J^{\prime} \parallel {\hat{T}}^{(E2)}\parallel J\rangle {| }^{2},\end{eqnarray}$
$\begin{eqnarray}B(M1,J\to J^{\prime} )=\displaystyle \frac{1}{2J+1}| \langle J^{\prime} \parallel {\hat{T}}^{(M1)}\parallel J\rangle {| }^{2},\end{eqnarray}$
where J and $J^{\prime} $ are the initial and final spin angular momenta, respectively. The numerical calculation of energy spectrum and wave functions can be accomplished by using the computer code NPBOS [53].

3. Results and discussion

In this study, we take the Z = 28 and N = 50 as closed shells. For ${}_{34}^{76}$Se42 nucleus, six valence protons construct Nπ = 3 particle-like proton bosons above the doubly magic core, and eight valence neutron holes construct Nν = 4 hole-like neutron bosons. By a least-square fitting to the experimental energies of 76Se nucleus and at the same time keeping a minimum number of free parameters, the obtained best IBM-2 parameters are listed as follows: ϵν = 1.41 MeV, ϵπ = 0.34 MeV, κπν = − 0.095 MeV, ${\chi }_{\pi }=-{\chi }_{\nu }=\sqrt{7}/2$, ωππ = 0.025 MeV, λ1 = 0.4 MeV, λ2 = 0.22 MeV, λ3 = − 0.2 MeV and ${c}_{\pi }^{(4)}=-0.08$ MeV for 76Se. In the following, we will discuss the low-lying energy structure and E2 and M1 transition strengths in comparison with the experimental data.

3.1. Energy levels

Firstly, we analyze the band structure. In figure 1, we compare the main calculated energy levels and E2 transitions with the new available experimental data. The other energy levels can be seen in table 1 which are useful for the discussion on the E2 and M1 transitions, the states of β band are shown in bold type. The most calculated results of the ground band, β band and γ band consists with the experimental data nicely. But the calculated ${3}_{1}^{+}$ state of γ band has a little deviations from the experimental value, resulting in the staggering in this band, which can be further solved by introducing other quadrupole–quadrupole interaction between like-bosons [38, 39]. The occurrence of low-lying ${0}_{2}^{+}$ state is intimately related with shape coexistence [21]. From figure 1, one can see that the excitation energy of ${0}_{2}^{+}$ state and the corresponding ${0}_{2}^{+}\to {2}_{1}^{+}$ E2 transition can be well reproduced in our calculation.
Figure 1. The calculated partial excitation energies (right panel) are compared with the experimental ones (left panel) for 76Se. The experimental data are taken from the [11, 28].
Table 1. The predicted levels are compared with the experimental results (taken from [11, 28]). The states of β band are shown in bold type. .
${E}_{\mathrm{Exp}}$(MeV) EIBM(MeV)
${0}_{1}^{+}$ 0.000 0.000
${2}_{1}^{+}$ 0.559 0.557
${0}_{2}^{+}$ 1.122 1.140
${2}_{2}^{+}$ 1.216 1.154
${4}_{1}^{+}$ 1.331 1.334
${3}_{1}^{+}$ 1.689 1.861
${{\bf{2}}}_{3}^{+}$ 1.788 1.875
${4}_{2}^{+}$ 2.026 2.043
${2}_{4}^{+}$ 2.127 2.029
${6}_{1}^{+}$ 2.263 2.350
${4}_{3}^{+}$ 2.485 2.364
${5}_{1}^{+}$ 2.489 2.566
${2}_{5}^{+}$ 2.515 2.383
${{\bf{4}}}_{4}^{+}$ 2.618 2.648
${2}_{6}^{+}$ 2.869 2.594
${1}_{1}^{+}$ 2.950 2.957
${2}_{7}^{+}$ 3.008 2.797
The U(5) and O(6) dynamical symmetries in IBM correspond to the spherical and γ-unstable nuclei, respectively [52]. Traditionally, one can differentiate the U(5) and O(6) characters by the structure of the spectrum. The most important signature of O(6) is the occurrence of nearly degenerate ${2}_{2}^{+}$ and ${4}_{1}^{+}$ states at about 2.5 times excitation energy of ${2}_{1}^{+}$ state and has a higher ${0}_{2}^{+}$ state. However, the level scheme of U(5) has the relation ${E}_{{2}_{2}^{+}}\approx {E}_{{0}_{2}^{+}}\approx {E}_{{4}_{1}^{+}}\approx 2{E}_{{2}_{1}^{+}}$ [42]. In table 1, our calculated result of two times ${E}_{{2}_{1}^{+}}$ is 1.114 MeV, the double experimental value of ${E}_{{2}_{1}^{+}}$ occurs at 1.118 MeV, both of which are close to ${0}_{2}^{+}$ and ${2}_{2}^{+}$ states. The calculated and experimental values for 2.5 times ${E}_{{2}_{1}^{+}}$ are 1.39 MeV and 1.40 MeV,respectively, which are far from the ${2}_{2}^{+}$ state at 1.216 MeV for experiment and 1.154 MeV for IBM-2 and ${4}_{1}^{+}$ state at 1.331 MeV for experiment and 1.334 MeV for IBM-2. Therefore, the low-lying energy level has a U(5)-like character.
In table 2, we display the two typical energy ratios. The energy ratio ${R}_{4/2}=E({4}_{1}^{+})/E({2}_{1}^{+})$ can be considered as a measure of quadrupole deformation. There are three characteristic values for different kinds of typical nuclei. For spherical nuclei near the magic core, there is R4/2 = 2.0. For γ-unstable nuclei, the typical value is R4/2 = 2.5. For rotational nuclei including mostly prolate shapes and less commonly oblate shapes, the ratio is R4/2 = 3.3 [22]. The experimental value of R4/2 for 76Se is 2.381, the calculation of IBM-2 gives R4/2 = 2.395, both of which indicate that 76Se is a transitional nucleus and intermediate between the U(5) limit and O(6) limit, the comparison is listed in table 2. Therefore, the ground state band of 76Se shows a γ-soft or triaxial shape. The energy ratio ${R}_{6/2}=E({6}_{1}^{+})/E({2}_{1}^{+})$ demonstrates similar behavior.
Table 2. The energy ratios R4/2 and R6/2 in the three dynamical limits are compared to the experimental and calculated values for 76Se. The three dynamical limits are taken from [30, 42].
${R}_{4/2}={E}_{{4}_{1}^{+}}/{E}_{{2}_{1}^{+}}$ ${R}_{6/2}={E}_{{6}_{1}^{+}}/{E}_{{2}_{1}^{+}}$
U(5) 2.0 3.0
SU(3) 3.3 7.0
O(6) 2.5 4.5
Exp. 2.381 4.048
IBM-2 2.395 4.219

3.2. E2 transition

In this section, we will discuss the properties of E2 transitions. Besides the energy spectrum, the transition strength B(E2) and some typical branching ratios can provide important information for the nuclear structure, because the absolute B(E2) values and some B(E2) ratios are highly sensitive to the different shapes of nuclei. The effective boson charges eρ in equation (2) are determined by fitting to the experimental data of $B(E2,{2}_{1}^{+}\to {0}_{1}^{+})=44(1)$ W.u. and $B(E2,{2}_{3}^{+}\to {0}_{1}^{+})=0.2(1)$ W.u. for 76Se nucleus. Then we obtain the eπ = 0.1492 eb and eν = 0.022eb, which are similar to the second set of effective charges in [54]. The neutron effective charge eν is substantially smaller than proton effective charge eπ, the physical reason might be attributed to the neutron subshell effect [33]. For the magnetic transitions, without loss of generality, we assume gν = 0 for the orbital effective g factor for the neutron bosons, and proton effective g factor gπ is determined to reproduce the experimental value of $B(M1,{1}_{1}^{+}\to {0}_{1}^{+})=0.010(1)$ ${\mu }_{N}^{2}$. The obtained gπ is 0.6 μN.
To see the detailed information about the E2 transitions, as shown in table 3, we carry out a comparison between experimental and calculated E2 and M1 transition rates of 76Se, the important E2 transitions are shown in bold type. Table 3 indicates that the overall agreement between the IBM-2 calculation and experimental data for B(E2) and B(M1) transition strengths is satisfactory, especially for strong electromagnetic transitions.
Table 3. Comparison between experimental and calculated E2 transition strengths (in W.u.) and M1 transition strengths (in ${\mu }_{N}^{2}$) for 76Se nucleus. The experimental data are taken from [11, 28].
${J}_{i}^{\pi }\,\to $ ${J}_{f}^{\pi }$ B(E2) B(M1)
Exp. IBM-2 Exp. IBM-2
${{\bf{0}}}_{2}^{+}\to {{\bf{2}}}_{1}^{+}$ 47(22) 42.40
${1}_{1}^{+}\to {0}_{1}^{+}$ 0.010(1) 0.010
${1}_{1}^{+}\to {2}_{1}^{+}$ ${0.04}_{1}^{5}$ 1.86 ${0.010}_{10}^{1}$ 0.0002
${2}_{1}^{+}\to {0}_{1}^{+}$ 44(1) 44.0
${2}_{2}^{+}\to {0}_{1}^{+}$ 1.21(10) 0.026
${{\bf{2}}}_{2}^{+}\to {{\bf{2}}}_{1}^{+}$ 43(3) 47.05 0.000 92(9) 0.0028
${2}_{3}^{+}\to {0}_{1}^{+}$ 0.2(1) 0.20
${{\bf{2}}}_{3}^{+}\to {{\bf{0}}}_{2}^{+}$ 37(10) 17.26
${2}_{3}^{+}\to {2}_{1}^{+}$ 1.1(3) 0.00 0.008(3) 0.0051
${{\bf{2}}}_{3}^{+}\to {{\bf{2}}}_{2}^{+}$ ${{\bf{4.5}}}_{16}^{35}$ 6.38 ${{\bf{0.009}}}_{5}^{4}$ 0.0129
${{\bf{2}}}_{3}^{+}\to {{\bf{4}}}_{1}^{+}$ 23(6) 11.40
${2}_{5}^{+}\to {0}_{2}^{+}$ 0.06(1) 0.46
${2}_{5}^{+}\to {2}_{1}^{+}$ 0.07(2) 0.26 0.0010(3) 0.0409
${2}_{5}^{+}\to {2}_{3}^{+}$ 4(1) 14.64 0.05(1) 0.0091
${2}_{5}^{+}\to {3}_{1}^{+}$ ${1}_{1}^{9}$ 4.74 0.0001(1) 0.0171
${2}_{6}^{+}\to {0}_{1}^{+}$ 0.23(2) 0.08
${2}_{6}^{+}\to {2}_{1}^{+}$ 0.68(8) 0.31 0.018(2) 0.0029
${2}_{6}^{+}\to {2}_{2}^{+}$ 1.1(1)a, $4.8{(}_{28}^{14}$)b 0.0005 0.027(3)a, $0.014{(}_{4}^{10})$ b 0.0148
${2}_{7}^{+}\to {0}_{1}^{+}$ 0.20(1) 0.12
${2}_{7}^{+}\to {2}_{1}^{+}$ 0.27(2) 0.49 0.083(8) 0.0090
${2}_{7}^{+}\to {2}_{2}^{+}$ 0.22(3)c, $5{(}_{3}^{88})$ d 0.13 0.021(3)c, 0.001(1)d 0.0169
${3}_{1}^{+}\to {2}_{1}^{+}$ 2.8(8) 0.37 0.0008(2) 0.0011
${{\bf{3}}}_{1}^{+}\to {{\bf{2}}}_{2}^{+}$ 18(5)e, ${\bf{1}}23{(}_{54}^{64})$ f 38.22 0.03(1)e, 0.001(1)f 0.0169
${{\bf{3}}}_{1}^{+}\to {{\bf{4}}}_{1}^{+}$ ${\bf{1}}8{(}_{8}^{35})$ 13.47 ${\bf{0.005}}{(}_{5}^{2})$ 0.0147
${{\bf{4}}}_{1}^{+}\to {{\bf{2}}}_{1}^{+}$ 71(2) 63.35
${4}_{2}^{+}\to {2}_{1}^{+}$ 0.08(2) 0.098
${{\bf{4}}}_{2}^{+}\to {{\bf{2}}}_{2}^{+}$ ${\bf{3}}{1}_{6}^{9}$ 29.50
${{\bf{4}}}_{2}^{+}\to {{\bf{4}}}_{1}^{+}$ ${\bf{2}}8{(}_{12}^{26})$ g, ${\bf{3.7}}{(}_{9}^{14})$ h 21.92 0.003(2)g, ${\bf{0.018}}{(}_{4}^{7})$ h 0.0153
${4}_{3}^{+}\to {2}_{2}^{+}$ 4.0(6) 0.27
${4}_{3}^{+}\to {3}_{1}^{+}$ ${1.3}_{2}^{3}$ 33.67 0.027(5) 0.055
${4}_{3}^{+}\to {4}_{1}^{+}$ 2.0(3) 0.11 ${0.028}_{4}^{5}$ 0.028
${{\bf{4}}}_{4}^{+}\to {{\bf{2}}}_{3}^{+}$ 31(5) 10.01
${{\bf{5}}}_{1}^{+}\to {{\bf{3}}}_{1}^{+}$ 67(23) 33.40
${5}_{1}^{+}\to {4}_{1}^{+}$ 4.7(17) 0.77 0.0006(4) 0.0043
${{\bf{6}}}_{1}^{+}\to {{\bf{4}}}_{1}^{+}$ 68(8) 61.39

a The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{0.38}_{12}^{14}$.

b The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{1.1}_{8}^{3}$.

c The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.21(19)$.

d The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{5}_{2}^{58}$.

e The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+0.41(5)$.

f The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+5.4(9)$.

g The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{2.4}_{6}^{10}$.

h The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.36(15)$.

The calculated E2 transition strength from ${0}_{2}^{+}$ to ${2}_{1}^{+}$ of $B(E2,{0}_{2}^{+}\to {2}_{1}^{+})=42.40$ W.u. nearly consists with the experimental value of 47(22)W.u.. The ${0}_{2}^{+}$ state can decay to ${2}_{1}^{+}$ state in U(5) limit, but this transition is forbidden in O(6) limit [42]. This is one of the most typical differences between U(5) and O(6) limit. The nonzero $B(E2,{0}_{2}^{+}\to {2}_{1}^{+})$ also implies that ${0}_{2}^{+}$ state is spherical. The E2 transition strength of $B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ is close to zero near the SU(3) limit and nonzero in the spherical and γ-soft shapes. The experimental and calculated values of $B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ are 43(3) and 47.05 W.u.. In any case, a transition towards the SU(3) limit seems to be ruled out completely. The interband E2 transition from ${2}_{2}^{+}$ to ${0}_{1}^{+}$ state is forbidden in all three limits of IBM, which is one of the most sensitive transitions for the recognition of shapes of nuclei. The calculation gives $B(E2,{2}_{2}^{+}\to {0}_{1}^{+})=0.026$ W.u., which is smaller than the experimental value of 1.21(10) W.u.. The E2 transition from ${3}_{1}^{+}$ to ${2}_{1}^{+}$ belonging to transition between γ band and ground state band is forbidden in the O(6) limit, the calculation and corresponding experimental data are 0.37 and 2.8(8) W.u., respectively. Both of the above two E2 transitions have a large experimental uncertainty. The calculated E2 transition strength of $B(E2,{4}_{2}^{+}\to {2}_{2}^{+})=29.50$ W.u. can nicely reproduce the experimental value of ${31}_{6}^{9}$ W.u.. The calculated intraband E2 transition from ${5}_{1}^{+}$ to ${3}_{1}^{+}$ in γ band has a strength of 33.40 W.u., which is a little smaller than experimental data 67(23) W.u.. In [2], their calculation for this E2 transition is 38 W.u. using IBM based on energy density functional. These E2 transitions indicate that these states of 76Se also has O(6) or γ-soft shape.
Some B(E2) branching ratios turn out to be powerful indicators of shape information. In this paper, we mainly concentrate on the four key sensitive quantities ${R}_{1}=B(E2,{4}_{1}^{+}\to {2}_{1}^{+})/B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$, ${R}_{2}=B(E2,{2}_{2}^{+}\to {2}_{1}^{+})/B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$, ${R}_{3}=B(E2,{0}_{2}^{+}\to {2}_{1}^{+})/B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$, and ${R}_{4}=B(E2,{2}_{2}^{+}\to {0}_{1}^{+})/B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ [33, 55]. The other useful B(E2) ratios will also be mentioned. Table 4 shows a comparison between IBM-2 calculation and the experimental data for these B(E2) ratios. These B(E2) ratios have different characteristic values of dynamical symmetry limits of IBM. We also list the values of R1, R2, R3 and R4 in the U(5), SU(3), and O(6) dynamical symmetry limits of IBM in table 4.
Table 4. Comparison of some B(E2) branching ratios in the three dynamical limits and the experimental and calculated values for 76Se. The values of three dynamical limits are taken from [33, 42, 46, 52, 5557].
$\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ $\tfrac{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ $\tfrac{B(E2,{0}_{2}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ $\tfrac{B(E2,{2}_{2}^{+}\to {0}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$ $\tfrac{B(E2,{6}_{1}^{+}\to {4}_{1}^{+})}{B(E2,{2}_{1}^{+}\to {0}_{1}^{+})}$ $\tfrac{B(E2,{4}_{1}^{+}\to {2}_{1}^{+})}{B(E2,{2}_{2}^{+}\to {2}_{1}^{+})}$
U(5) 2.0 2 2 0.011 3.0 1.0
SU(3) 10/7 0.0 0 0.70 1.57 6.93
O(6) 10/7 10/7 0 0.07 1.67 1.84
Exp. 1.614 0.977 1.07 0.028 1.545 1.65
IBM-2 1.440 1.069 0.964 0.0006 1.395 1.346
$\tfrac{B(E2,{4}_{2}^{+}\to {4}_{1}^{+})}{B(E2,{4}_{2}^{+}\to {2}_{2}^{+})}$
U(5) 0.72
SU(3) 0.03
O(6) 0.75
Exp. 0.90a, 0.12b
IBM-2 0.743

a The corresponding multipole mixing ratio ${\delta }_{-}^{+}=+{2.4}_{6}^{10}$.

b The corresponding multipole mixing ratio ${\delta }_{-}^{+}=-0.36(15)$.

The calculated R1 values essentially reproduce the experimental data. The experimental ratio R1 = 1.614 appears to be in between the U(5) limit (R1 = 2.0) and O(6) limit (R1 = 10/7). The calculated R1 value of 1.440 is a little closer to the O(6) limit relative to U(5) limit, which suggests a rather γ-soft feature comparing to experimental value. The ratio R2 is a sensitive quantity to detect the shape characteristics of nuclei. The calculated R2 = 1.069 fairly consists with the experimental value of 0.977, both of which are close to the O(6) limit(R2 = 10/7), ruling out the SU(3) limit(R2 = 0). Since the E2 transition strength of $B(E2,{0}_{2}^{+}\to {2}_{1}^{+})$ is as strong as the ${2}_{1}^{+}\to {0}_{1}^{+}$ transition, the calculated ratio R3 = 0.964 which is very close to the observed value of 1.07 nearly lies at the middle point of the U(5) (R3 = 2.0) and O(6) limit (R3 = 0). In this case, we cannot distinguish whether the 76Se nucleus belongs to spherical shape or γ-soft shape. As for R4, there exists the large errorbar in the transition $B(E2,{2}_{2}^{+}\to {0}_{1}^{+})$, the experimental value R4 = 0.028 has a little larger discrepancy and has much spherical character. The calculated value of R4 of 0.0006 is far less than the corresponding experimental data. Both the experimental and calculated R4 is much closer to the value 0.011 of U(5) limit.
In addition, we compare the other four B(E2) ratios as shown in table 4. The calculated and experimental ratios of $B(E2,{6}_{1}^{+}\to {4}_{1}^{+})$/$B(E2,{2}_{1}^{+}\to {0}_{1}^{+})$ demonstrate that 76Se is more close to O(6) limit. The calculation and experimental result of $B(E2,{4}_{1}^{+}\to {2}_{1}^{+})$/$B(E2,{2}_{2}^{+}\to {2}_{1}^{+})$ which are intermediate between the U(5) and O(6) limits have a little discrepancy, our result is 1.346, which is close to the U(5) limit, the experimental data is 1.65, which is more close to O(6) limit. As for the ratio of $B(E2,{4}_{2}^{+}\to {4}_{1}^{+})$/$B(E2,{4}_{2}^{+}\to {2}_{2}^{+})$, there is a large uncertainty for different multipole mixing ratios, which are not good structure indicators now, the further experimental data are needed for these ratio.
From the above discussion, both the experimental and theoretical results of the key sensitive quantities have confirmed that there is a coexistence between spherical and γ-soft shapes in 76Se nucleus. Microscopically, similar to the Z = 40 subshell effect on 76Kr, the shape coexistence in 76Se nucleus also might originate from effect of the N = 40 subshell closure [58], the more analysis from the theoretical study is needed.

3.3. M1 transition.

All the experimental M1 transitions in 76Se are less than 0.1 ${\mu }_{N}^{2}$ or even more smaller. Table 3 shows the calculated M1 transition strengths of 76Se in comparison with available experimental data. One can see that most of the calculated and experimental results are in good agreement for the M1 transition probabilities within experimental uncertainty both quantitatively and qualitatively.
For the strong M1 transitions from ${4}_{3}^{+}$ to ${3}_{1}^{+}$ and from ${4}_{3}^{+}$ to ${4}_{1}^{+}$, our calculated results are 0.055 and 0.028 ${\mu }_{N}^{2}$, respectively, which can nicely reproduce the experimental values of 0.027(5) and ${0.028}_{4}^{5}$ ${\mu }_{N}^{2}$. The ${1}_{1}^{+}$ state is known as the scissors mode. The experimental data give that M1 transition strength from ${1}_{1}^{+}$ to ${0}_{1}^{+}$ is $B(M1,{1}_{1}^{+}\to {0}_{1}^{+})=0.010(1)$ ${\mu }_{N}^{2}$, which is used to fit the parameter gπ. The M1 transition strength of $B(M1,{1}_{1}^{+}\to {0}_{1}^{+})$ is 0 and 0.079 ${\mu }_{N}^{2}$ in the U(5) and O(6) limits, respectively [50, 51]. This indicates that it is close to U(5) limit.
From table 3, one can find that the M1 transition strength of $B(M1,{4}_{2}^{+}\to {4}_{1}^{+})$ is very weak, which belongs to M1 transition from quasi-γ band to the ground state band, the experimental value is 0.003(2)(or ${0.018}_{4}^{7}$) ${\mu }_{N}^{2}$, the predicted result is 0.0153 ${\mu }_{N}^{2}$. From the viewpoint of IBM-2, the asymmetry between the parameters χπ and χν which brings MS components into low-lying states results in the sizeable M1 transition strength from ${4}_{2}^{+}$ to ${4}_{1}^{+}$. In our calculation, the parameters χρ satisfy the relation χπ + χν = 0, which brings about the situation similar to O(6) limit [59, 60].
The calculated result for M1 transition from ${2}_{2}^{+}$ to ${2}_{1}^{+}$ is $B(M1,{2}_{2}^{+}\to {2}_{1}^{+})=0.0028$ ${\mu }_{N}^{2}$, which quantitatively agrees with the corresponding experimental value of 0.000 92(9) ${\mu }_{N}^{2}$. The weak M1 transition implies that both ${2}_{2}^{+}$ and ${2}_{1}^{+}$ are symmetric states. The similar analysis can be applied to the M1 transitions from ${2}_{3}^{+}$ to ${2}_{1}^{+}$ (or ${2}_{2}^{+}$) states. The experimental data are well reproduced by the theoretical prediction. The calculated M1 transition strength of $B(M1,{3}_{1}^{+}\to {2}_{1}^{+})$ is 0.0011 ${\mu }_{N}^{2}$, which is consistent with the experimental value of 0.0008(2)${\mu }_{N}^{2}$. The calculated transition strengths of $B(M1,{3}_{1}^{+}\to {2}_{2}^{+})$ and $B(M1,{3}_{1}^{+}\to {4}_{1}^{+})$ are in good agreement with experimental values, but these M1 transitions are also very weak. The further experimental investigation is needed to determine the corresponding multipole mixing ratio. The M1 decay transition from ${2}_{5}^{+}$ to ${2}_{3}^{+}$ is underestimated by a factor of five in our calculation, possibly due to the restricted model space.

4. Conclusion

In summary, we have investigated the low-lying states and electromagnetic transitions in 76Se in the framework of IBM-2. Our calculation can well reproduce the the properties of ${0}_{2}^{+}$ state, the calculated energy ratios R4/2 and R6/2 agree with the corresponding experimental data. We also calculate the several B(E2) branching ratios such as R1, R2, R3 and R4, the calculations can nicely describe the experimental results. In the meantime, we compare the theoretical M1 transitions with the recent experiment. Based on the analysis on the energy energy ratios, the B(E2) branching ratios and M1 transitions, we can conclude that there is a coexistence between spherical shape and γ-soft shape in 76Se.

The authors are grateful to Professor Y X Liu and Professor G L Long for helpful discussions. This work was supported by the National Natural Science Foundation of China under Grant Nos. 12047568 and 11147148 and the Natural Science Foundation of Zhejiang Province under Grant No. LY19A050002.

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