-
The study on the nuclei around the
132 Sn core is a crucial topic in current research [1-3]. The neutron-rich nuclei in the132 Sn region lie away from the line of stability and are reported to be essential in understanding the r-process nucleosynthesis, as well as the evolution of nuclear deformation [4-11]. With the continuous development of new facilities, it has become possible to experimentally study the low-lying energy spectrum and electromagnetic properties of a few nuclei in this region. This also provides theorists with a new vision for the study on the neutron-rich nuclei.The long even-even Te isotope chain, which has two protons above the
Z=50 shell closure, plays an important role in studying the evolution of the nuclear structure. Experimentally, the low-lying energies of Te isotopes with neutron numbers from 54 to 88 have been measured [12]. Except for the nuclei nearN=82 , the energy ratio of the first4+ to2+ state is approximately equal to 2, thereby indicating a vibration structure [13, 14]. However, theE2 transitions of a few light Te isotopes do not follow that of a typical vibrator [15-17]. The low-lying energy levels withN>82 exhibit different characteristics. Both the first2+ and4+ energies of the nuclei withN>82 are lower than those withN<82 . This suggests that the nuclei withN>82 are more collective [18]. In contrast, the ratio ofE(6+1)/E(2+1) inN>82 is generally greater than that inN<82 , which is propably owing to the significant contribution of neutrons in theνf7/2 orbital, coupled with protons in theπg7/2 orbital [18].The measurement of electromagnetic properties in this nuclear region is a challenging task. To date, the heaviest Te nucleus with known reduced transition probabilities and magnetic moments is
136 Te withN=84 [19-23]. However, there is a noticeable variety in magnitudes forB(E2;0+1→2+1) transitions [B(E2)↑ for short] of136 Te obtained in different experiments. The first measurement [19, 20] suggested that theB(E2)↑ value of136 Te [0.122(18) e2 b2 ] was significantly lower than that of132 Te [0.216(22) e2 b2 ], thereby indicating asymmetry with respect to theN=82 shell closure. This result was understood as a neutron dominance in the neutron-proton exchange symmetry, generally preserved in most Te isotopes withN<82 , and broken for the nucleus136 Te [24-26]. Two of the latest experiments provided different results, i.e.,B(E2)↑=0.181(15) e2 b2 in Refs. [21, 22] and0.191(26) e2 b2 in Ref. [23]. In addition, the g factor evolution symmetry was verified via experiments [21, 22]. The resultant g factor of136 Te has a similar magnitude to that of132 Te, which suggests that the2+1 state of136 Te is not as extreme a neutron-dominant as had been thought [21, 22].The objective of this study is to investigate the low-lying band structures and electromagnetic properties of even-even Te isotopes with mass numbers from 128 to 140 via the nucleon pair approximation (NPA) of the shell model [27-30]. Recently, the NPA has been successfully applied to study low-lying states, vibration-rotation phase transitions, and regularities of nuclear structures under random interactions in the transitional region [31-48]. It has also been generalized with isospin symmetry [49], particle-hole excitations [50], m-scheme basis [51, 52], and deformation [53-55]. For a more comprehensive review, refer to Ref. [30]. The shell model configurations of the nuclei studied in this research are constructed by valence proton particles, valence neutron holes (
N<82 ), and valence neutron particles (N>82 ) with respect to132 Sn, a doubly closed nucleus. We diagonalize a phenomenological shell model Hamiltonian in a collective pair subspace, which is an approximation of the full shell model space for low-lying states. In Fig. 1, the space dimensions of the Te isotopes between the full shell model and the NPA model space adopted in this study are compared. It can be inferred that the dimension of the SM configurations varies from the single digit atN=82 to∼106 atN=88 . The dimension of the NPA truncated space is significantly smaller, which allows us to evaluate the dominant configuration in a straightforward manner, thus providing us with a simple and clear picture of the nuclear structure.Figure 1. (color online) Comparison of the space dimensions for the Te isotopes between the full shell model and the NPA truncated space adopted in this study. The shell model configuration space is constructed by using valence protons outside the
Z=50 closed shell and valence neutrons with respect to theN=82 closed shell.This paper is organized as follows. In Sec. II, we provide a brief introduction to the NPA, including the Hamiltonian, model space, and transition operators. In Sec. III, we present our results obtained from the calculations on the energy levels of low-lying states,
B(E2) transition rates between these levels, and g factors. The summary and conclusions of the study are provided in Sec. IV. -
In this study, we introduce the phenomenological Hamiltonian to describe Te isotopes, which has been wildly adopted in previous NPA calculations [31-43]. It includes the spherical single-particle energy term
H0 , a residual interaction containing the monopole pairingHP0 , the quadrupole pairingHP2 between similar nucleons, and the quadrupole-quadrupole interactionHQ between all valence nucleons :H=H0+HP0+HP2+HQ,
(1) with
H0=∑jσϵjσC†jσCjσ,HP0=∑σ−G0σP(0)†σ⋅P(0)σ,HP2=∑σ−G2σP(2)†σ⋅P(2)σ,HQ=∑σ−κσQσ⋅Qσ+κπνQπ⋅Qν.
where
C†jσ andCjσ are single-particle creation and annihilation operators, respectively.σ=π,ν refers to the proton and neutron degrees of freedom.ϵjσ indicates the single-particle energy. The pairing and quadrupole operators are defined by:P(0)†σ=∑jσ√2jσ+12(C†jσ×C†jσ)(0)0,P(2)†σ=∑jσj′σq(jσj′σ)(C†jσ×C†j′σ)(2)M,Qσ=∑jσj′σq(jσj′σ)(C†jσטCj′σ)(2)M.
where
q(jj′)=Δjj′(−)l+l′+1(−)j−12ˆj^j′√20πC20j12,j′−12⟨nl|r2|nl′⟩ , withΔjj′=12[1+(−)l+l′+2] , andC20j12,j′−12 is the Clebsch-Gordan coefficient.G0σ ,G2σ ,κσ , andκπν represent the two-body interaction strengths corresponding to the monopole, quadrupole pairing, and quadrupole-quadrupole interactions between all valence nucleons.The single-particle energies and two-body interaction parameters corresponding to the proton and neutron excitations in our calculations are shown in Tables 1 and 2, respectively. The nuclei with neutron number
N<82 are treated in terms of valence neutron holes, while those withN>82 are considered in terms of valence neutron particles. In Table 1, the proton single-particle energiesd3/2 ,d5/2 ,g7/2 , andh11/2 are obtained from the experimental excitation energies of133 Sb [12]. There are no experimental data available for the remaining orbitals1/2 , and we obtain its single-particle energy from a shell-model study in the132 Sn region [56]. The neutron hole-like and particle-like single-particle energies are extracted from the corresponding experimental excitation energies [12] of131 Sn and133 Sn, respectively. The adopted two-body interaction parametersG0σ ,G2σ ,κσ , andκπν in Table 2 are obtained by fitting the experimentally excited energies and electromagnetic properties of low-lying states.j s1/2 d3/2 d5/2 g7/2 h11/2 ϵjπ 2.990 2.440 0.962 0.000 2.792 N<82 j s1/2 d3/2 d5/2 g7/2 h11/2 ϵjν 0.332 0.000 1.655 2.434 0.065 N>82 j p1/2 p3/2 f5/2 f7/2 h9/2 i13/2 ϵjν 1.363 0.8537 2.0046 0.000 1.5609 2.690 Table 1. Adopted single-particle energies
ϵjπ andϵjν (in MeV) for Te isotopes (Z=52 ) withN<82 andN>82 , respectively.N⩽82 G0ν G2ν κν G0π G2π κπ κπν 0.17 0.021 0.04 0.18 0.018 0.039 +0.08 N>82 G0ν G2ν κν G0π G2π κπ κπν 0.11 0.01 0.02 0.13 0.024 0.05 −0.08 Table 2. Adopted two-body interaction parameters
G0σ ,G2σ ,κσ , andκπν .σ=π,ν stands for proton and neutron, respectively.G0σ is in unit of MeV;G2σ ,κσ , andκπν are in MeV/r40 withr20=1.012A13 fm2 .The model space in the NPA is constructed by collective nucleon-pairs, defined by
A(r)†σ=∑jσj′σy(jσj′σr)(C†jσ×C†j′σ)(r)M,
with
r=0,2,4,6,8 corresponding to S, D, G, I and/or K pairs, respectively.y(jσj′σr) is called the structure coefficient of the spin-r pair. Based on the single-particle energies in Table 1 and our experience, the model space we selected is as follows. For the proton degree of freedom, collectiveSπ ,Dπ ,Gπ , andIπ pairs are taken to construct the proton nucleon-pair basis. For the neutron degree of freedom, collectiveSν,Dν,Kν pairs [orSν,Dν,Gν,Iν , andKν pairs] are included in the neutron-hole [or neutron-particle] pair basis.The
E2 transition operator is defined byT(E2)=eπQπ+ eνQν, whereeπ andeν correspond to the effective charges of valence proton and valence neutron. TheB(E2) value in units ofW.u. is given byB(E2;Ji→Jf)=2Jf+12Ji+1×(eπχπ+eνχν)2r405.94×10−6×A4/3,
(2) with reduced matrix element
χσ=⟨βf,Jf||Qσ||βi,Ji⟩ (σ=π,ν ) andr20=1.012A1/3 fm2 .|βi,Ji⟩ is the eigenfunction carrying angular momentumJi and the symbolβi represents all quantum numbers other thanJi . Our proton effective charge is taken to beeπ=1.6e , the same as in previous calculations in this mass region [5, 31]. The neutron effective chargeeν=−1.20e forN<82 andeν=0.74e forN>82 are obtained by fitting to the experimental data.The
M1 transition operator is given byT(M1)= √34π∑σ=π,νglσLσ+gsσSσ , whereLσ andSσ represent the orbital and spin angular momenta, respectively.glσ andgsσ correspond to the orbital and spin gyromagnetic ratios, respectively. The magnetic dipole moment is defined byμ(Ji)=√4π3CJiJiJiJi,10(ξlπ+ξsπ+ξlν+ξsν)μN,
(3) with reduced matrix elements
ξlσ=⟨βi,Ji||glσLσ||βi,Ji⟩ andξsσ=⟨βi,Ji||gsσSσ||βi,Ji⟩ (σ=π,ν ). Here,μN is the nuclear magneton. The g factor is defined byμ(Ji)/μNJi , and it is expressed asg(Ji)=√4π3CJiJiJiJi,10Ji(ξlπ+ξsπ+ξlν+ξsν).
(4) In the above unit convention, the g factors,
Lσ ,Sσ , andJi are dimensionless. The effective spin gyromagnetic ratios are taken to begsπ=5.586×0.7 andgsν=−3.826×0.7 . Namely, the quenching factor 0.7 is adopted. According to previous theoretical calculations in this region [21, 22, 57, 58], the orbital gyromagnetic ratio of protonglπ=1.1 is adopted in this study. The orbital gyromagnetic ratios of the neutron areglν=0.025 forN<82 , andglν=0.189 forN>82 , determined by theχ2 fitting of experimental g factors. -
In this section, we present the results obtained from our calculations on even-even Te isotopes with neutron numbers from 76 to 88. We discuss the low-lying structures of these nuclei and focus on the symmetric and asymmetric structural evolutions with neutron numbers. The calculated level schemes are compared with experimental ones, as presented in Fig. 2; the comparisons of
B(E2) values and g factors are presented in Table 3. As can be observed, our results agree well with the experiment, especially for the yrast states. This indicates that the NPA provides us with an appropriate theoretical framework to study low-lying states of the Te isotopes, below and above theN=82 shell closure.Figure 2. Low-lying states in even-even nuclei
128−132 Te (N<82 ) and136−140 Te (N>82 ). Experimental data are obtained from Ref. [12]. Experimental levels with "()" correspond to cases for which the spin and/or parity of the corresponding states are not well established.Nuclei State B(E2) g(Jπi) Jπi Jπf Exp. Cal. Exp. Cal. 128 Te2+1 0+1 19.68(18) 21.3 +0.318(13) +0.29 4+1 2+1 18.4 +0.63 6+1 4+1 9.7(6) 7.7 +0.76 8+1 6+1 <0.001 −0.22 130 Te2+1 0+1 15.1(3) 16.3 +0.351(18) +0.33 4+1 2+1 11.9 +0.70 6+1 4+1 6.1(3) 5.9 +0.77 8+1 6+1 <0.001 −0.22 132 Te2+1 0+1 10(1) 10.7 +0.28(15)(+)0.38(4) +0.41 (+)0.46(5) 4+1 2+1 6.6 +0.75 6+1 4+1 3.3(2) 3.9 +0.79(9) +0.78 8+1 6+1 <0.001 −0.22 134 Te2+1 0+1 6.3(20) 4.6 +0.76(9) +0.81 4+1 2+1 4.3(4) 4.2 +0.70+0.55−0.38 +0.79 6+1 4+1 2.05(4) 2.3 +0.846(25) +0.79 136 Te2+1 0+1 7.6(15) 8.9 (+)0.34(+8−6) +0.40 4+1 2+1 14.1(21) 9.9 +0.27 6+1 4+1 4.1 −0.04 8+1 6+1 5.6 +0.26 138 Te2+1 0+1 11.5 +0.28 4+1 2+1 13.1 +0.14 6+1 4+1 9.2 +0.22 8+1 6+1 9.4 +0.41 140 Te2+1 0+1 13.4 +0.27 4+1 2+1 18.1 +0.34 6+1 4+1 0.6 −0.14 8+1 6+1 0.4 +0.53 -
First, we look at the low-lying energy levels. The energy ratio of the first
4+ to2+ state, namelyR4/2=E(4+1)/E(2+1) , is a well known indicator of nuclear collectivity [59]. In Fig. 2, the experimentalR4/2 ratio is approximately2.01 forN=76,88 ,1.95 forN=78,86 , and1.71 forN=80,84 . This indicates that the energy ratiosR4/2 of Te isotopes exhibit a symmetric pattern with respect to theN=82 shell closure [13, 14]. From the values ofR4/2 , both128−132 Te withN<82 and136−140 Te withN>82 exhibit vibration-like characters.The
E(2+1) energy is also a widely used structural indicator besidesR4/2 . The experimental data and the corresponding NPA results versus the neutron number N are plotted in Fig. 3(a). From the general trend,E(2+1) energies show obvious asymmetric characteristics with respect to theN=82 shell closure. The measured values in theN>82 region are lower than those in theN<82 region. This was explained as a reduced neutron pairing above theN=82 shell [24]. To verify whether or not our calculation agrees with this explanation, we further investigate the contribution of different Hamiltonian interactions [i.e.,H0 ,HP0 ,HP2 , andHQ in Eq. (1)] to theE(2+1) energies presented in Fig. 3(b). It can be observed thatHP0 dominates the evolution ofE(2+1) . As illustrated in Fig. 3(c), both neutronHP0ν and protonHP0π vary greatly after crossingN=82 . Therefore, the drop ofHP0 asN>82 results from the combination of both proton and neutron monopole pairings. We also note that the monopole pairing strength, i.e.,G0σ herein, in theN>82 region is weaker than that in theN<82 region. Therefore, we suggest that this asymmetric feature ofE(2+1) energies is primarily owing to the evolution of residual monopole pairing interactions acrossN=82 , which is similar to the argument in Ref. [24].Figure 3. (color online) (a)
E(2+1) energies (in MeV), (b) contributions of different Hamiltonian interactions (denoted asH0 ,HP0 ,HP2 , andHQ ), and (c) contributions of neutronHP0ν and protonHP0π .H0 is the spherical single-particle energy term.HP0 andHP2 refer to the residual monopole pairing and quadrupole pairing interactions between similar nucleons, respectively.HQ includes the quadrupole-quadrupole interactions between all valence nucleons. Our results (cal.), which include all aforementioned Hamiltonian terms, are shown as blue squares. The experimental data (exp.) are obtained from Ref. [12]. -
Nuclear electromagnetic properties, such as
B(E2) transitions and g factors, are sensitive probes for detecting quadrupole collectivity and deformation. Before the first measurement of theg(2+1) factor of136 Te [21, 22], the available experimentalB(E2;2+1→0+1) transition of136 Te was approximately half that of132 Te [19, 20]. This was theoretically interpreted as the neutron dominance in the2+1 -state wave function [24-26], and a negativeg(2+1) factor of136 Te [24, 25] was predicted as a consequence of the overestimation of the contribution of neutron excitations to the total wave function [68]. Experimentally, the sign ofg(2+1) factor is suggested to be positive [(+)0.34(+8−6) ], thus indicating that the2+1 -state wave function of136 Te is not completely dominated by the neutron configuration as previously suggested [21, 22]. To elucidate the above discussions in the literature, in this subsection, we would like to focus on the evolution trends ofB(E2) transitions and g factors, as well as the dominant configuration of the2+1 state. The experimental data and the corresponding NPA results as a function of N are presented in Fig. 4(a) and 5(a), respectively. It can be inferred that our calculated results generally agree well with the experimental data.Figure 4. (color online) (a)
B(E2;2+1→0+1) values (in W.u.) and (b) the matrix elements of protoneπχπ and neutroneνχν . Our results with adopted effective charges (eν=−1.20e forN<82 andeν=0.74e forN>82 ) and symmetric effective charges (eν=−e forN<82 andeν=e forN>82 ) are represented by blue squares and black asterisks, respectively. The experimental data are obtained from Refs. [12, 19-23].As illustrated in Fig. 4(a), the measured
B(E2;2+1→0+1) transitions exhibit certain asymmetric patterns around theN=82 shell. Although theB(E2) data of136 Te (N=84 ) vary slightly based on experiments, they are generally smaller than that of132 Te (N=80 ). We further study the contributions of protons and neutrons to theB(E2) values, as presented in Fig. 4(b). From Eq. (2),B(E2;2+1→0+1)=C[eπχπ+eνχν]2 with the coefficientC=0.2×r405.94×10−6×A4/3 . Because C varies slightly in theA∼132 region, it can be approximately taken as a constant (0.13 fm4 ). Therefore the contribution of protons and neutrons toB(E2) values only depends on the matrix elementseπχπ andeνχν . It can be inferred that the evolution trend of neutron matrix elements (eνχν -adopted) in Fig. 4(b) is approximately the same as that ofB(E2) values (Cal. with adoptedeν ) in Fig. 4(a). We thus suggest that the asymmetric evolution trend ofB(E2) with N is primarily determined by the neutron part.To understand why the neutron part (
eνχν -adopted) is asymmetric with respect to theN=82 shell closure, we carefully consider the dominant configurations of2+1 states. For brevity, we omit S pairs and abbreviate the NPA basis|(D†π)nπ(S†π)Nπ−nπ(D†ν)nν(S†ν)Nν−nν⟩ to be|(Dπ)nπ(Dν)nν⟩ . For example,|Dν⟩ represents the NPA basis|(S†π)NπD†ν(S†ν)Nν−1⟩ . Our results are as follows. For the nuclei withN<82 ,|2+1;132Te⟩:0.68|Dν⟩,0.70|Dπ⟩,|2+1;130Te⟩:0.72|Dν⟩,0.64|Dπ⟩,|2+1;128Te⟩:0.72|Dν⟩,0.60|Dπ⟩.
And for the nuclei with
N>82 ,|2+1;136Te⟩:0.68|Dν⟩,0.69|Dπ⟩,|2+1;138Te⟩:0.71|Dν⟩,0.54|Dπ⟩,|2+1;140Te⟩:0.70|Dν⟩,0.50|DνGν⟩,0.52|Dπ⟩.
Here, the coefficient in front of each configuration is the projection of the basis and the wave function, i.e.,
⟨basis|2+1⟩ . The configurations with(projection)2 less than 20% are omitted. It can be observed that the2+1 state of most Te nuclei (except for140 Te) is dominated by mixing|Dπ⟩ and|Dν⟩ . Hence, the dominant configuration of the wave function can be approximated as symmetric at approximatelyN=82 . In contrast,140 Te has one more component|DνGν⟩ in its dominant configuration. This suggests that the configuration-mixing is enhanced in this nucleus. In the literature,138 Te is located in the transitional region [69], and the shape transition from spherical in136 Te to prolate in140 Te is predicted to take place at139 Te [4].Symmetric wave functions lead to symmetric
E2 matrix elementsχν . Therefore, the asymmetry of theB(E2) has to be traced back to the neutron effective chargeeν . To highlight the asymmetry of the adoptedeν , we reintroduce the symmetric effective charges witheν=−e forN<82 andeν=e forN>82 . Our results are denoted by "Cal. with symmetriceν '' in Fig. 4(a) and "eνχν -symmetric'' in Fig. 4(b). Significantly large deviations from experimental data can be observed, if symmetriceν is assumed. Therefore we suggest that the asymmetric pattern ofB(E2) is related to the asymmetry of neutron effective charges (i.e., different core polarization characters below and aboveN=82 ).Unlike
B(E2) transitions andE(2+1) energies, theg(2+1) factors in Fig. 5(a) are nearly symmetric. According to Eq. (4),g(2+1)≃0.84(ξlπ+ξsπ+ξlν+ξsν) . In other words, theg(2+1) factor comprises proton orbital, proton spin, neutron orbital, and neutron spin parts. We calculate these four parts and present their corresponding reduced matrix elements in Fig. 5(b). One sees that the evolution trend ofg(2+1) factors is primarily determined by the proton orbital partξlπ . The contribution of the neutron orbital partξlν is close to zero, while the contributions of the proton/neutron-spin parts (ξsπ andξsν ) are negative.To this point, we have separately discussed the general trends of
B(E2;2+1→0+1) transitions in Fig. 4 andg(2+1) factors in Fig. 5. Next, we discuss these two simultaneously. It can be observed in Fig. 4(b) that the matrix elements of the proton and neutron contribute with the same phase in the evaluation of theE2 transition; hence, the precision of the non-dominate component is not crucial. In the corresponding calculation ofg(2+1) values in Fig. 5(b), the proton/neutron orbital and spin components contribute out of phase and may lead to significant errors if they are not precisely evaluated. Therefore, the g factor is very sensitive to the details of the wave function, and in particular, the balance between proton and neutron contributes to the wave function. As we have analyzed before, the different evolution behaviors of theB(E2) transitions and g factors regardingN=82 are primarily owing to the different roles played by the neutrons and protons. Becuase the proton component dominates the evolution of g factors, the basic symmetric distribution of g factors withN=82 indicates that the proton contribution in the2+1 -state wave function of136 Te should be equivalent to that of132 Te. In other words, the2+1 -state wave function of136 Te is actually not absolutely dominated by the neutron configuration, as mentioned in Refs. [21, 22]. -
Another interesting asymmetrical behavior is related to the energy level spacing between the
4+1 and6+1 states [E(6+1)−E(4+1) ]. Our obtained results are presented in Fig. 6. It can be observed that the spacing of nuclei withN<82 is generally smaller (the experimental values are approximately 0.10 MeV atN=80 , 0.18 MeV atN=78 , and 0.31 MeV atN=76 ). After crossing theN=82 shell, this level spacing increases rapidly to 0.35 MeV atN=84 and then continues to 0.54 MeV atN=86 . Possibly, owing to the emphasis on the collectivity of our calculation, there is a slightly systematic overestimation ofE(4+1) in Fig. 6(a) and a corresponding underestimation of [E(6+1)−E(4+1) ] in Fig. 6(b).Figure 6. (color online) (a)
E(4+1) andE(6+1) energies and (b) the energy level spacing of adjacent4+1 and6+1 states [E(6+1)−E(4+1) ] (in MeV). The experimental data are obtained from Ref. [12].The asymmetrical pattern of [
E(6+1)−E(4+1) ] can be explained in part by the dominant configurations of the4+1 and6+1 states. For the nuclei withN<82 ,|4+1;132Te⟩:0.93|Gπ⟩;|4+1;130Te⟩:0.85|Gπ⟩;|4+1;128Te⟩:0.77|Gπ⟩,0.51|DπDν⟩;|6+1;132Te⟩:0.96|Iπ⟩;|6+1;130Te⟩:0.92|Iπ⟩;|6+1;128Te⟩:0.88|Iπ⟩.
It can be deduced that the
4+1 (and6+1 ) states of132 Te and130 Te are very pure proton excitations dominated by|Gπ⟩ (and|Iπ⟩ ) in the NPA basis. This causes the energy levels of the two nuclei to be significantly close. As presented in Table 1, theπg7/2 orbit is the lowest proton single-particle energy level. The two-valence-proton configuration(πg7/2)2 accounts for 99% in the collectiveGπ andIπ pairs. Hence, the small spacing between4+1 and6+1 states belowN=82 is mainly caused by the spin alignment of two protons in theπg7/2 orbit. For128 Te, although its6+1 state is also dominated by|Iπ⟩ , its4+1 state is mixed with another configuration|DπDν⟩ , so the4+1 state is suppressed, and the spacing increases. For the nuclei withN>82 , their dominant configurations are represented as|4+1;136Te⟩:0.56|Gν⟩,0.66|DπDν⟩;|4+1;138Te⟩:0.47|D2ν⟩,0.57|DνIν⟩,0.54|DπDν⟩;|4+1;140Te⟩:0.50|D2ν⟩,0.62|DπDν⟩,0.46|DπDνGν⟩;|6+1;136Te⟩:0.81|Iν⟩;|6+1;138Te⟩:0.57|Iν⟩;|6+1;140Te⟩:0.70|Iν⟩,0.73|DνIν⟩.
It can be deduced that the dominant configuration of the
4+1 state relatively differs from that of the6+1 state. Because the configuration mixing of4+1 state is significantly more complicated in theN>82 region, the4+1 state is suppressed lower, which in turn causes the energy level spacing [E(6+1)−E(4+1) ] to increase. -
In summary, we calculated low-lying level schemes,
B(E2) transitions, and g factors of even-even Te isotopes with neutron numbers from 76 to 88 via the nucleon-pair approximation (NPA). The optimal agreement with experiments indicates that our theoretical framework is suitable for studying low-lying structures of the nuclei in this neutron-rich mass region.We compared the yrast band structures of
N<82 andN>82 nuclei. The energy ratio of the4+1 to2+1 state is symmetric at approximatelyN=82 , and its value varies at approximately2.0 . However, theE(2+1) energies in theN>82 region are lower than those in theN<82 region. This corresponds to a reduction in the contribution of residual monopole pairing interactions between similar nucleons in theN>82 region. However, a small energy spacing between the4+1 and6+1 state appears in nuclei withN<82 , and disappears in nuclei withN>82 . We analyzed the dominant configurations of the4+1 and6+1 states. Such an asymmetric behavior ofE(6+1)−E(4+1) is mainly caused by the spin alignment of two protons in theπg7/2 orbit of theN<82 region.We also studied the evolution trends of
B(E2;2+1→0+1) transitions andg(2+1) factors with the neutron number N. We inferred that the asymmetric behavior ofB(E2) with respect toN=82 is primarily determined by the neutron contribution, which indicates different core polarization characters, below and above theN=82 shell. In contrast,g(2+1) factors vary symmetrically at approximatelyN=82 . This pattern was determined to be dominated by the proton-orbit part. Furthermore, in Table 3, we presented the theoretical predictions ofE2 transitions and g factors for a few yrast states, which are still experimentally unknown. We expect our prediction to be beneficial in future studies on these nuclei.
Symmetric and asymmetric structural evolutions of Te isotopes across the N = 82 shell closure
- Received Date: 2021-04-28
- Available Online: 2021-09-15
Abstract: Systematic calculations of low-lying energy levels,