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The characterization of the underlying structure of 124Te is difficult, and it has attracted a significant amount of attention in various theoretical and experimental studies [1-6]. The 124Te lies outside the major shell
Z=50 with two protons, and its ratioE(4+1/2+1)=2.07 is very close to the spherical vibrator value of 2.00. This indicates that the 124Te is a vibrational or U(5) symmetric nucleus. However, the energy levels spread of the two-phonon multiplets do not suggest a vibrational character in 124Te. Meanwhile, the energy level of the lowest excited 0+ state suggests that the 124Te exhibits moreγ -unstable or O(6) behavior; however, theE2 decays of the 0+2 state do not support theγ -soft rotational nature.On the contrary, shape coexistence in the Te isotopes is still elusive [4]. The appearance of strong
ρ(E0,0+2→0+1) transition is an important spectroscopic fingerprint that mostly supports the shape coexistence in nucleus [7]. For 124Te, the experimental value of the electric monopole transition strengthρ2(E0,0+2→0+1)×103 is12±3 [8]. This value is very similar to the corresponding transition in the Cd isotopes; the latter demonstrates firm evidence for the nature of shape coexistence, providing strong support for the deformed intruder structure of the first-excited 0+ state in 124Te isotopes. However, it does not provide conclusive proof for the shape coexistence in 124Te yet [4]. Meanwhile, systematic research on the known experimental information suggests that 124Te may be an example of the nucleus at theE(5) critical point [9]. Recent investigations indicate that 124Te exhibitsE(5) -like structure [5, 10]. However, the low-lying structure of 124Te has still not been understood (see, e.g., Refs. [3, 10, 11] and references therein).The
E(5) andX(5) symmetries originally developed by Iachello [12, 13] correspond to an exact solution of the Bohr Hamiltonian forγ -independent potentials and an approximate solution of the Bohr-Hamiltonian forγ≈0∘ , respectively. From the Bohr-Hamiltonian withγ frozen atγ=30∘ , a four dimensional critical point symmetry model calledZ(4) was introduced [14]. Most recently, another four-dimensional critical point symmetry calledT(4) was obtained from the Bohr-Hamiltonian with theβ -soft potential and for a fixed value ofγ with0∘⩽γ⩽30∘ [15].In the nuclear system, the
E(5) symmetry can be used to describe a nucleus at the critical point of the second-order phase transition from a spherical vibrator to aγ -soft rotor. In the interacting boson model (IBM-1) [16], which does not distinguish between the proton boson and neutron boson, theE(5) represents a nucleus that is located at the critical point of the shape phase transition between the U(5) and O(6) symmetries. Because the neutron and proton degrees of freedom are explicitly taken into consideration, the proton-neutron interacting boson model (IBM-2)[16] has a complex phase diagram [17, 18]. The critical point of the second-order phase transition can occur at the phase transition fromUπν(5) toOπν(6) symmetry and can be subsumed to the critical point of the second-order phase transition in IBM-1 [17-19]. Compared to their neighboring isotopes, nuclei at or close to the critical point of shape phase transition demonstrate drastic changes in the properties of the low-lying states, such as the energy levels, theE2 transition strengths, andE2 branching ratios [20]. However, so far, there has been no discussion on the critical point symmetry in the real nuclei for the phase transition betweenUπν(5) andOπν(6) in the IBM-2 phase diagram.The low-lying levels of 124Te have recently been examined with the (
n,n′,γ ) reaction. The spins, level energies,B(E2) transition probabilities, and multipole-mixing ratios were obtained [3]. The experiment provides the basis for an in-depth study of the low-lying structure of 124Te. In this work, we study the low-lying structure of the 124Te within the framework of the IBM-2, with a focus on the key observables of the critical point of the second-order transition betweenUπν(5) andOπν(6) . We also calculate the low-lying energy levels and theE2 ,M1 , andE0 transition strengths, and we compare the predictions of the critical point symmetry of theUπν(5) -Oπν(6) transition with the experimental data. We attempt to describe the critical point symmetry in the 124Te, and reveal the low-lying structure of 124Te in IBM-2 space.The outline of the paper is as follows. In Section 2, we introduce the Hamiltonians of IBM-2, E2, M1, and E0 operators, used in this work. The criteria adopted for the determination of the model parameters and the comparison of the numerical results with experimental data and the predictions of the critical point symmetry are presented in Section 3. Finally, the conclusions and summary are provided in Section 4.
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The IBM-2 is an algebraic model, in which valence nucleons are coupled to form
sρ bosons (angular momentumL =0) anddρ bosons (angular momentumL=2 ), whereρ=π denotes proton bosons andρ=ν denotes neutron bosons, respectively. The microscopic structure of the model suggests that there are only two significant terms: the pairing interaction between identical nucleons and the quadrupole-quadrupole interaction between nonidentical nucleons. The simple standard IBM-2 Hamiltonian [16, 21] is written asˆH=εdπˆndπ+εdνˆndν+κπνˆQπ⋅ˆQν,
(1) where
ˆndρ=d†ρ⋅~dρ andˆQρ=(s†ρ˜dρ+d†ρsρ)(2)+χρ(d†ρ˜dρ)(2) representd -boson number operator and quadrupole operator, respectively. The parameterχρ determines the type of the deformation of the quadrupole operator.εdρ is the energy of the d bosons relative to thes bosons, andκπν is the strength of the quadrupole-quadrupole interaction between the neutron boson and the proton boson.The Hamiltonian of Eq. (1) has a much richer shape phase structure, which contains
Uπν(5) ,Oπν(6) ,SUπν(3) ,¯SUπν(3) , andSU∗πν(3) dynamical symmetries, corresponding to the spherical vibrator,γ -unstable rotor, axially symmetric prolate rotor, axially symmetric oblate rotor, and triaxial rotor, respectively. The shape phase transitions in nuclei can be characterized as the quantum phase transitions between the different dynamical symmetries in the IBM [17, 18, 22]. Although the standard IBM-2 Hamiltonian can provide a clearer space of dynamical symmetry, other physical dominant interactions must be included to describe the real nuclei more accurately. We use the following IBM-2 Hamiltonian in this study [16, 23, 24],ˆH=εdπˆndπ+εdνˆndν+κπνˆQπ⋅ˆQν+ωπνˆLπ⋅ˆLν+ˆMπν,
(2) where
^Lρ=√10[d†ρ⋅~dρ](1) is the angular momentum operator with a dipole proton-neutron interaction parameterωπν , andˆMπν=λ2(s†πd†ν−s†νd†π)(2)⋅(sπ˜dν−sν˜dπ)(2)+∑k=1,3 λk(d†πd†ν)(k)⋅ (˜dπ˜dν)(k) is the Majorana interaction; the Majorana parametersλk (k= 1, 2, 3) represent the strength of the Majorana interaction. If the Majorana term is not included in Eq. (2), the adopted parameters in the remaining four terms of Eq. (2) indicate that the nucleus described by the IBM-2 may be located on the the plane ofUπν(5) -Oπν(6) -SU∗πν(3) , because theˆLπ⋅ˆLν only generates the physical rotational groupSOπν(3) of the IBM-2 [16-18]. On the contrary, in Ref. [25] the authors proposed a new algebraic modelF(5) based on the Euclidean dynamical symmetry in five dimensions (Eu(5) ), which can build a symmetry intermediate between theE(5) andX(5) symmetries [22]. However, theF(5) cannot directly be defined in the IBM-1 or IBM-2 because of the noncompactness of theEu(5) group [25].The decay properties of the low-lying states, such as the
B(E2) transition probabilities, the magnetic dipoleB(M1) transition strengths and theρ2(E0) values between the lowest 0+ states are the characteristics of the phase structure. In the IBM-2, the E2 transition strength is given by the following expressionB(E2,J′→J)=12J′+1|J⟨‖
(3) where
J' andJ are the angular momenta for the initial and final states, respectively. The\hat{T}{(E2)}=e_{\pi} \hat{Q}_{\pi} + e_{\nu}\hat{Q}_{\nu} is the E2 operator, where the operatorQ_\rho is the same as in Eq. (1). The parameterse_\pi ande_\nu are the effective charges of proton bosons and neutron bosons, respectively. The values ofe_\nu ande_\pi could be taken differently.In the proton-neutron interacting boson model, the magnetic dipole
M1 transition operator is defined as\hat T\left( {M1} \right) = \sqrt {\frac{3}{{4\pi }}} \left( {{g_\nu }{{\hat L}_\nu } + {g_\pi }{{\hat L}_\pi }} \right),
(4) where the
\hat{L}_\rho is the same as in Eq. (2). Theg_\pi andg_\nu are the effective proton and neutron bosong -factors, respectively. Typically, one can take the values ofg_\pi=1 ,g_\nu=0 in the calculations [21].The
E0 transition matrix element\rho in the IBM-2 is written as [26, 27]\begin{split}\rho \left( {E0,J' \to J} \right) = & \frac{Z}{{e{R^2}}}\left[{\beta _{0\pi }}\langle J\left\| {{{\hat T}_\pi }\left( {E0} \right)} \right\|J'\rangle \right. \\ & \left.+ {\beta _{0\nu }}\langle \left\| {{{\hat T}_\nu }\left( {E0} \right)} \right\|JJ'\rangle \right]\end{split},
(5) where
R=1.2A^{1/3} fm,\beta_{0\pi} , and\beta_{0\nu} are the effective monopole charges of proton and neutron boson in units ofe fm2, respectively. The E0 operator is expressed as\hat T\left( {E0} \right) = {\beta _{0\pi }}{\hat T_\pi }\left( {E0} \right) + {\beta _{0\nu }}{\hat T_\nu }\left( {E0} \right) = {\beta _{0\pi }}{\hat n_{d\pi }} + {\beta _{0\nu }}{\hat n_{d\nu }} , where the\hat{n}_{d\rho} is the same as in Eq. (1). -
The 124Te is located between the vibrational nucleus and the deformed nucleus [28]. The IBM provides a powerful tool to describe the nuclear shapes and the shape phase transitions. Several studies have used this approach to describe the properties of the low-lying states of 124Te isotope. In IBM-1, early studies regarded the 124Te as an example of O
(6) symmetry; however, the energy levels of yrast states of 124Te differ from the O(6) limit [29]. Further systematic investigations found that 124Te was close to the U(5) symmetry [30, 31]. Moreover, recent results of experiments and calculations indicate that 124Te may possess theE(5) features [3, 9, 5, 10]. In IBM-2, the calculations with and without the mixed configuration suggest that 124Te is an O(6) -like nucleus, but theB(E2) decay pattern is not consistent with this symmetry [32]. A subsequent study showed that neither O(6) nor U(5) symmetries can describe the 124Te well [33], although the potential energy surface of 124Te exhibits an obvious manifestation of O(6) limit [34]. Another recent experiment and IBM calculation support that 124Te demonstrates soft triaxial behavior [6]. Although the low-lying structure of the 124Te is still not understood, it is indubitable that these results indicate that 124Te is a transitional nucleus between vibration U(5) symmetry and\gamma -unstable O(6) symmetry in IBM, and might accompany with a slight soft triaxial rotation.To describe the low-lying structure of 124Te, we selected the doubly closed shell
^{132}_{50} Sn_{82} as the inert core. There areN_\pi=1 particle-like bosons beyond theZ=50 major shell, andN_\nu=5 hole-like neutron bosons from theN=82 shell closure; the total number of bosons isN_B=6 . Considering that the proton bosons are particle-like and the neutron bosons are hole-like based on different single particle orbitals in 124Te, we choose different relative energies ford neutron and proton boson in this case, i.e.,\varepsilon_{d\pi}\neq\varepsilon_{d\nu} , similar to Refs. [35-37]. To describe the different type of valence neutrons and protons exhibiting opposite intrinsic quadrupole deformation, and to obtain the\gamma -unstable O(6) symmetry, we adopt\chi_\pi+\chi_\nu=0 . To introduce some characteristics of the soft triaxial rotation in 124Te, the value of\chi_\pi and\chi_\nu should not be zero. In principle, the parameter\varepsilon_{d\rho} mainly leads to vibrational solutions, while\kappa_{\pi\nu} drives the system into a deformed shape, and the description of the electromagnetic properties is very sensitive to the parameters\chi_\rho . The dipole interaction\hat{L}_{\pi}\cdot\hat{L}_{\nu} can improve the rotational spectrum and can adjust the position of the 4^+_1 relative to 2^+_2 state; however, it does not affect the wave vectors [38]. The Majoranan operator mainly contributes to the mixed symmetry states including the scissors mode. All the free parameters in our calculations were fixed to reproduce the experimental energies and electromagnetic transition strengths of 124Te. By fitting to the energy levels of the experimental data, especially for the 2^+_1 state, we obtain the parameters of IBM-2 in this study as follows:\varepsilon_{d\pi}=1.220 MeV,\varepsilon_{d\nu}=0.710 MeV,\kappa_{\pi\nu}=-0.140 MeV,\chi_\pi=-\chi_\nu=-1.00 ,\omega_{\pi\nu}=-0.055 MeV,\lambda_1=-0.700 MeV,\lambda_2=0.220 MeV, and\lambda_3=-0.100 MeV. The IBM-2 Hamiltonian will be numerically diagonalized by using the NPBOS code [39]. In Fig. 1, we compare the calculated level energies with the experimental results [3, 40]. The experimental level energies in Fig. 1 are separated into different bands based only on their order of appearance; only the low-spin positive parity states with uniquely assigned spin and parity are displayed in the figure.Figure 1 shows that the theoretical levels are in good agreement with the corresponding experimental levels. In particular, the 6
^+_1 state is dominated by two valence protons and is relatively unaffected by the increase of neutrons. The 6^+_1 states remain at a nearly constant excitation energy for even-even 122-130Te isotopes [3]. By introducing the dipole interaction term into the Hamiltonian [41-43], the structure of the yrast bands including the 6^+_1 level are reproduced by the theoretical predictions. Similar to the 6^+_1 state, the observed energy of 4^+_2 level is only changed by approximately 135 keV from 122Te to 130Te isotopes. The present calculation reproduces the experimental energy of the 4^+_2 level well. Meanwhile, the present calculated energies of the 2^+_{3, 4, 5} , and 6^+_2 levels agree with the experimental observations. Even though the 0^+_2 level is clearly higher than that is expected for a two-phonon multiple state, the description of the first two excitation 0^+ states in energy sequence is satisfactory. Furthermore, the calculated energy level of the first scissor mode 1^+_1 perfectly reproduces the experimental data. However, there are some discrepancies between the experimental observations and the theoretical predictions for higher energy states, such as 2^+_6 , and 2^+_7 states, which is a general feature of this model [44]. For the 0^+_4 state, the calculated level is significantly higher than the experimental level. The IBM-2 with the mixed configuration shows that 0^+_4 is an intruder state in 124Te and is located outside the IBM-2 space [32].In order to investigate the properties of the
E2 transitions in 124Te, the effective charges of the proton and neutron bosons were determined to reproduce the experimentalB(E2) values. Using the same method as in Refs. [45, 46] and exactly fitting to the experimental data ofB(E2, 2^+_1\rightarrow0^+_1)=31(5) W.u., we obtaine_\pi=3.780 \sqrt{\rm {W.u.}} ande_\nu=1.670 \sqrt{\rm {W.u.}} . The theoretical calculation ofB(E2) values for 124Te in comparison with the available experimental data [1, 3, 10, 47-49] is given in Table 1.{ J^\pi_i\rightarrow } { J^\pi_f } Expt. IBM-2 { { 2^+_1 }\rightarrow { 0^+_1 }} {31^5_5 } 31.02 { { 4^+_1 }\rightarrow { 2^+_1 }} 54.51 48.06 {{ 2^+_2 }\rightarrow { 2^+_1 }} { 55.5^{109}_{99} } 49.53 {{ 2^+_2 }\rightarrow { 0^+_1 }} {0.83 ^{23}_{16} } 0.24 {{ 0^+_2 }\rightarrow { 2^+_1 }} { 20^{4}_{4} } 27.92 {{ 0^+_3 }\rightarrow { 2^+_2 }} 50.00 48.51 {{ 4^+_2 }\rightarrow { 2^+_2 }} {14.1 ^{30}_{28} } 29.94 { { 4^+_2 }\rightarrow { 4^+_1 }} {12.7 ^{74}_{59} } 27.06 { { 4^+_2 }\rightarrow { 2^+_1 }} {4.3 ^{9}_{9} } 0.31 {{ 3^+_1 }\rightarrow { 2^+_2 }} { 59^{10}_{10} } 42.65 { { 2^+_3 }\rightarrow { 0^+_1 }} { 0.26^{4}_{4} } 2.08 {{ 2^+_3 }\rightarrow { 2^+_1 }} {0.025 ^{4}_{3} } 0.50 {{ 2^+_3 }\rightarrow{ 2^+_2 }} {\leqslant 2.7^{19}_{15} } 4.03 { { 2^+_3 }\rightarrow { 4^+_1 }} {0.81 ^{104}_{81} } 5.38 { { 2^+_3 } \rightarrow { 3^+_1 }} {0.26 ^{5}_{4} } 0.37 { { 2^+_4 }\rightarrow{ 2^+_2 }} {0.29 ^{39}_{33} } 1.85 { { 2^+_4 }\rightarrow{ 2^+_1 }} {1.6 ^{3}_{3} } 1.39 { { 2^+_4 }\rightarrow{ 0^+_1 }} { 0.053^{17}_{14} } 2.16 {{ 0^+_4 }\rightarrow{ 2^+_1 }} { <0.5 } 0.04 { { 0^+_4 }\rightarrow{ 2^+_2 }} { <50 } 0.03 { { 1^+_1 }\rightarrow{ 2^+_1 }} {1.2 ^{4}_{3} } 4.99 { { 1^+_1 }\rightarrow{ 2^+_2 }} { 0.74^{38}_{38} } 1.13 {{ 4^+_3 }\rightarrow { 2^+_2 }} {13.4 ^{49}_{39} } 1.16 {{ 4^+_3 }\rightarrow{ 4^+_1 }} { 6.7^{23}_{20} } 2.89 {{ 4^+_3 }\rightarrow{ 2^+_1 }} { 1.9^{6}_{5} } 0.95 {{ 2^+_5 }\rightarrow{ 2^+_2 }} {0.61 ^{40}_{30} } 1.02 {{ 2^+_5 }\rightarrow { 2^+_1 }} {0.06 ^{7}_{6} } 0.15 { { 2^+_5 }\rightarrow{ 0^+_1 }} { 0.078^{50}_{42} } 0.09 {{ 2^+_6 }\rightarrow{ 2^+_2 }} { 1.6^{5}_{5} } 1.67 { { 2^+_6 }\rightarrow{ 4^+_1 }} { 1.7^{5}_{4} } 0.00 {{ 2^+_6 }\rightarrow{ 2^+_1 }} {0.046 ^{8}_{6} } 0.23 {{ 2^+_6 }\rightarrow{ 0^+_1 }} {0.12 ^{3}_{2} } 0.01 {{ 4^+_4 } \rightarrow{ 4^+_1 }} { 2.6^{9}_{7} } 0.10 { { 4^+_4 }\rightarrow{ 2^+_1 }} { 0.051^{28}_{22} } 0.72 {{ 2^+_7 }\rightarrow{ 2^+_1 }} {4.0 ^{4}_{4} } 0.56 {{ 2^+_7 }\rightarrow{ 2^+_2 }} {0.20 ^{10}_{11} } 0.00 { { 2^+_7 }\rightarrow{ 4^+_1 }} { <1.6 } 1.10 Table 1 shows that the computed
{B}(E2) transition strengths are in overall agreement with the experimental data, although most cases of the experimental value have a large uncertainty. In particular, the typical strongly collectiveE2 transitions with tens of Weisskopf units are described by the theoretical predictions very well, some of them consist with each other within the experimental uncertainty. It is noteworthy that the experimental{B}(E2, 4^+_1\rightarrow2^+_1) transition strength is35.9(\pm 17) W.u. in Ref. [3], which is taken from Ref. [6]. In Ref. [48], the experimental values show that the lower and upper limits of this transition probability are 27.3 and 163.5 W.u., respectively. Similar to Ref. [47], we adopt the experimental averaged value 54.51 W.u. for this transition; the details are provided in Ref. [50]. The experimental value agrees with the present calculated value. At the same time, the experimental relatively strongB(E2) transitions, which are comparable with the experimental transition probability of the first-excited state 2^+_1 decay to the ground state with dozens of W.u., are reproduced by the calculated results perfectly except the{B}(E2, 4^+_3\rightarrow2^+_2) transition strength. Furthermore, the calculated results are in good description of the properties of the experimental weakly collectiveE2 transitions with about one or even less than one W.u.. The theoreticalE2 transition strengths from the scissor mode 1^+_1 to the 2^+_1 and 2^+_2 states are in agreement with the corresponding experimental data, although the theoretical values are slightly higher than the experimental data testTo identify where the 124Te can be placed in the
\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) transition, we focus on a set of key observables [51], such as the energy ratiosR_{4_1/2_1}=E(4^+_1)/E(2^+_1) ,R_{2_2/2_1}=E(2^+_2)/E(2^+_1) ,R_{0_2/2_1}=E(0^+_2)/E(2^+_1) ,R_{0_3/0_2}=E(0^+_3)/ E(0^+_2) , and theB(E2) ratiosR_{B, 42}=B(E2, 4^+_1\rightarrow2^+_1)/ B(E2, 2^+_1\rightarrow0^+_1) ,R_{B, 22}=B(E2, 2^+_2\rightarrow2^+_1)/B(E2, 2^+_1\rightarrow0^+_1) andR_{B, 02}=B(E2, 0^+_2\rightarrow2^+_1)/B(E2, 2^+_1\rightarrow0^+_1) , which reflect the characteristics of the nucleus behavior at critical point of the phase transition from\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) in IBM-2 space [17]. These observables are the most crucial nuclear structure indicators [52]. Some indicators can even distinguish the first-order quantum phase transition from the second-order quantum phase transition [53-55]. The values of these key observables at the critical point of the phase transition from\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) for infinite numbers of bosons [12], as well as the experimental data and the calculated results in 124Te, are listed in Table 2 for comparison.CPST 124Te IBM-2 {R_{4_1/2_1} } 2.20 2.07 2.06 {R_{2_2/2_1} } 2.20 2.20 2.13 {R_{0_2/2_1} } 3.03 2.75 2.35 {R_{0_3/0_2} } 1.18 1.14 1.46 {R_{B, 42} } 1.68 1.74 1.55 {R_{B, 22} } 1.68 1.79 1.60 {R_{B, 02} } 0.86 0.65 0.90 Table 2. Comparison of the key observables of the states at the critical point of the
{\rm{U}_{\pi\nu}(5)} -{\rm{O}_{\pi \nu}(6)} transition (taken from Ref. [12] and denoted as CPST), the experimental data of 124Te (taken from Refs. [1, 3, 47-49] and labeled as 124Te) and the calculated results (labeled as IBM-2).Table 2 shows that the overall agreement is well. On the energy ratios, the experimental ratio
R_{2_2/2_1} is almost exactly reproduced to the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. The experimental ratioR_{0_3/0_2} is very close to the value of CPST. Meanwhile, the experimental energy of the0^+_2 state, relative toE(2^+_1) , satisfactorily matches the predicted value of CPST. The experimentalR_{4_1/2_1} lies between U_{\pi\nu}(5) (R_{2_2/2_1}=2.00 ) and\rm{O}_{\pi \nu}(6) (R_{2_2/2_1}=2.50 ) limits, although it deviates slightly from the predicted one. On theB(E2) ratios, the experimentalR_{B, 42} is well reproduced by the theoretical prediction and clearly points to a structure intermediate between U_{\pi\nu}(5) (R_{B, 42}=2.00 ) and\rm{O}_{\pi \nu}(6) (R_{B, 42}=10/7 ). Meanwhile, the experimental value ofR_{B, 22} also approaches the predicted value of the critical point theory. In particular, the information ofR_{B, 02} is a signature for identifying the\rm{U}_{\pi \nu}(5) from the\rm{O}_{\pi \nu}(6) symmetry. In\rm{U}_{\pi \nu}(5) limit, theB(E2, 0^+_2\rightarrow2^+_1) transition strength is two times larger than(B(E2, 2^+_1\rightarrow0^+_1) . In\rm{O}_{\pi \nu}(6) limit, theB(E2, 0^+_2\rightarrow2^+_1) transition is forbidden [56]. Here, both the experimental and theoretical values of theR_{B, 02} can correctly reflect the nature of theB(E2, 0^+_2\rightarrow2^+_1) transition, although the calculated value overestimates the experimental value by approximately 1.5 times. Therefore, all the available experimental information on the key observables for 124Te is in good agreement with the predictions of critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. Meanwhile, Table 2 also shows that the present calculation of the characteristic feature of 124Te is remarkable. However, theR_{0_2/2_1} of CPST seems larger than the IBM's in Table 2; this is because the calculation of the CPST is given for the infinite-N_B limit [12, 51]. For a finite number of bosons, Ref. [17] showed that theR_{0_2/2_1} of a nucleus at the critical point of the phase transition from\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) withN_{\pi} = N_{\nu} =5 is 2.48, which agrees the result of the present calculations. From the above discussion, we conclude that the 124Te may be a nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. Consequently, the0^+_2 state is the lowest possible state of the first-excited family of intrinsic levels predicted by the critical point symmetry.An advantage of IBM-2 over IBM-1 is that the former can study the influence of the critical point symmetry on magnetic dipole transitions between the low-lying states [23]. In IBM-2, the pure
\rm{O}_{\pi\nu}(6) symmetry can be obtained only with\chi_\pi=\chi_\nu=0 . In this case, there does not exist any asymmetry between\chi_\pi and\chi_\nu . This implies that the mixed symmetry components could not mix into the low-lying states. Therefore, theM1 transitions between these states are forbidden in\rm{O}_{\pi\nu}(6) limit [21]. The theoretical predictions show that theB(M1) transition strengths among the low-lying states almost vanish when a nucleus is at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition [17]. For the scissors mode state 1^+_1 , theB(M1, 1^+_1\rightarrow 0^+_1) transition strength is3N_\pi N_\nu(g_\nu-g_\pi)^2/{4\pi(2N_B+1)} for\rm{O}_{\pi \nu}(6) limit; however, it is forbidden for\rm{U}_{\pi\nu}(5) limit. TheB(M1, 1^+_1\rightarrow 0^+_2) value is3N_\pi N_\nu(g_\nu-g_\pi)^2/{\pi N_B(N_B-1)} for\rm{U}_{\pi\nu}(5) limit, whereas it vanishes for\rm{O}_{\pi\nu}(6) limit [23, 57, 58]. At the critical point of\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition, the 1^+_1 state allowsM1 decays to first and second0^+ states. In the case of 124Te, theB(M1, 1^+_1\rightarrow 0^+_1) is 0.10\mu^2_N for\rm{O}_{\pi \nu}(6) limit, andB(M1, 1^+_1\rightarrow 0^+_2) is 0.50\mu^2_N for\rm{U}_{\pi\nu}(5) limit, with the typical valuesg_\pi=1 andg_\nu=0 . Indeed, only a fewM1 transitions, with very small absolute values ofB(M1) , among the low-lying states in 124Te have been measured [49, 59]. By using Eq. (4) and also takingg_\pi=1 andg_\nu=0 , the calculatedM1 transition probabilities, as well as the corresponding available experimental data for the scissors mode and the low-lying states in 124Te, are provided in Table 3.From Table 3, it can be seen that all the experimental
B(M1) transition strengths between the lowest 2^+ states are of the order of approximately 0.01\mu^2_N or even less, which is far from the typicalB(M1) value of the mixed symmetry state. This demonstrates that the observedM1 transitions are in qualitative agreement with the predictions of the critical point symmetry of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. At the same time, Table 3 shows that the calculated results reasonably reproduce the characteristics of theM1 transitions among low-lying states. In particular, the calculatedB(M1, 1^+_1\rightarrow 0^+_1) strength is close to the experimental one, which is comparable to the predicted value of the critical point between the\rm{U}_{\pi \nu}(5) and\rm{O}_{\pi \nu}(6) limits. This indicates that both the experimental and the calculated results of theB(M1, 1^+_1\rightarrow 0^+_1) transition reflect the property of the nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. Unfortunately, theB(M1, 1^+_1\rightarrow 0^+_2) strength in 124Te has not been measured, the calculated value for this transition is 0.035\mu^2_N , locating between the\rm{U}_{\pi \nu}(5) and\rm{O}_{\pi \nu}(6) transition similar to the case ofB(M1, 1^+_1\rightarrow 0^+_1) transition. All these results indicate that both the calculated and experimentalM1 transitions are consistent with the characteristic behavior of the 124Te nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition.However, some deviations between the theoretical and experimental
M1 transition strengths are also are displayed in Table 3; all the calculatedM1 values are slightly larger than the corresponding experimental values. In IBM-2,\chi_\pi=-\chi_\nu\neq0 leads to a spectrum with many O(6) features. In addition, some mixed symmetry components from the asymmetry of\chi_\pi and\chi_\nu mix into the low-lying states beside the pure mixed symmetry states at higher energies [21, 61]. On the contrary,\chi_\pi=-\chi_\nu\neq0 makes the nucleus have some features of the\rm{SU}^\ast_{\pi \nu}(3) symmetry, because the proton and neutron have opposite types of deformation [18]. The calculatedB(M1) values among the lowest states are systematically slightly larger than the experimental data, which implies that the description of 124Te with the characteristics of the\rm{SU}^\ast_{\pi \nu}(3) is slightly rather than the actual nucleus. Furthermore, the smallB(M1) values among the low-lying states in 124Te also indicate that 124Te may contain some components of\rm{SU}^\ast_{\pi \nu}(3) symmetry. Therefore, the 124Te may be the nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition and with a slight\rm{SU}^\ast_{\pi \nu}(3) symmetry.Transition Expt. IBM-2 {2^+_2 } {\rightarrow } {2^+_1 } 0.0003 0.006 {2^+_3 } {\rightarrow } {2^+_2 } 0.004 0.025 {2^+_3 } {\rightarrow } {2^+_1 } 0.013 0.072 {2^+_4 } {\rightarrow } {2^+_2 } 0.001 0.031 {2^+_4 } {\rightarrow } {2^+_1 } 0.004 0.075 {1^+_1 } {\rightarrow } {0^+_1 } 0.04 0.029 {1^+_1 } {\rightarrow } {0^+_2 } 0.034 The electric monopole transition strengths between 0+ states can be considered as a characteristic of shape coexistence [7] and quantum phase transitions [62], and it can reflect the properties of
E(5) dynamical symmetry [9]. In fact, the\rho^2(E0, 0^+_2\rightarrow0^+_1) value of 124Te has been measured [8]. However, the\rho^2(E0) value on the first-excited 0+ state in 124Te has not been understood yet, as mentioned in the introduction [4]. In IBM, theE0 transitions do not occur in the U(5) dynamical limit, becauseE0 operator is proportional to\hat{n}_d [63].While in O(6) limit, the selection rules require theE0 transition from the 0^+_2 to ground state to be forbidden [64]. For the predictedE(5) symmetry, the 0^+_2 level, which is the lowest member (zero phonon) of the first-excited family, should have an allowedE0 branch to the 0^+_1 state. However, so far, the studies of the behavior ofE(5) symmetry in 124Te have not provided a detailed analysis of the properties ofE0 transitions. In IBM-2 space, all the symmetrical states have correspondence with the IBM-1 states; therefore, IBM-2 subsumes the critical point of second-order transition of IBM-1[17, 23]. The critical point symmetry of\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) has considerably similarity to the predictions of the{E}(5) symmetry [17]. Subsequently, the0^+_2 level should also have an allowedE0 decay to the ground state in this prediction of the critical point symmetry. We take the values of parameters\beta_{0\nu} and\beta_{0\pi} as in Ref. [65], namely,\beta_{0\nu}=0.25 and\beta_{0\pi}=0.10 e fm2. The calculated\rho^2(E0, 0^+_2\rightarrow0^+_1)\times10^3 value is11.00 , which is consistent with the experimental data within the experimental uncertainty. The theoretical and experimentalE0 transitions prove that the 0^+_2 state may be interpreted as the lowest state of the first-excited family of intrinsic levels in the critical point symmetry of second order\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition.It is well known that the low-lying yrast states in other Te isotopes such as 112, 114Te show similar behavior, i.e., typical vibrational like spectra, but other properties deviate from the vibrator's [6, 66-68]. These Te isotopes are located around the midshell 118Te. The large-scale shell-model calculations indicate that the behavior of the
B(E2) values of these nuclei is related to the competition between the seniority coupling and the neutron-proton correlations. From the viewpoint of the IBM model, 112Te is below the midshell 118Te; it hasN_\nu=5 bosons beyondN=50 major shell. Both 112Te and 124Te have the same numbers of bosons withN_\pi=1 andN_\nu=5 ; therefore, they show similar properties of spectrum. As for 114Te, the measured ratio ofB(E2, 4^+_1\rightarrow 2^+_1) toB(E2, 2^+_1\rightarrow 0^+_1) is less than one, which is a very unusual deformation; no theoretical models can provide a satisfactory description [68]. -
In summary, the low-lying structure of the 124Te was investigated within the framework of IBM-2. The calculated low-lying energy levels are in good agreement with the corresponding experimental data. In particular, the 6
^+_1 , 4^+_2 levels, which are maintain a nearly constant excitation energy from 122Te to 130Te isotopes, have been satisfactorily reproduced by using the calculated results. In addition, the first two excited 0+ states were described well in the energy sequence, although the 0^+_2 level is clearly higher than that is expected for a two-phonon multiple state. Furthermore, the calculated energy level of the first scissor mode 1^+_1 perfectly reproduces the experimental data.The calculations show that the computed
{B}(E2) transition strengths are in the overall agreement with the experimental data. The observed typical strongly collectiveE2 transitions with tens of Weisskopf units, and most of the experimental relatively strongB(E2) transitions comparable to the experimental{B}(E2, 2^+_1\rightarrow0^+_1) transition probability with dozens of W.u., are reproduced by the theoretical predictions nicely. Furthermore, the calculated results provide accurate description of the properties of the experimentally weakly collectiveE2 transition with approximately one or even less than one W.u.. Particularly, the theoreticalE2 transitions from the scissor mode 1^+_1 to the 2^+_1 and 2^+_2 states are in good agreement with the corresponding experimental data, although the theoretical values are slightly higher than the experimental data.By comparing the key observables of the states at the critical point of the U
_{\pi\nu}(5) -\rm{O}_{\pi \nu}(6) transition with the experimental data and the IBM-2 calculations, we show that both the experimental data and the theoretical results of the 124Te agree with the predicted properties of the states at the critical point of the phase transition from\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) very well. This indicates that the 124Te may be a nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. Consequently, the0^+_2 state is the lowest possible state of the first-excited family of the intrinsic levels predicted by the critical point symmetry.At the same time, the characteristics of the
B(M1) transition strengths among the low-lying states are consistent with the predictions of the critical point of the transition from\rm{U}_{\pi \nu}(5) to\rm{O}_{\pi \nu}(6) , and there is good agreement between the experiment and the theory for characteristics of theB(M1) transition strengths. Especially, both the experimental and theoreticalB(M1, 1^+_1\rightarrow 0^+_1) values are located between the\rm{U}_{\pi \nu}(5) and\rm{O}_{\pi \nu}(6) limits. By analyzing the calculated results and the structure parameters of the quadrupole operators in this work, all quantities support that the 124Te may be the nucleus at the critical point of the\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition and with feature of\rm{SU}^\ast_{\pi \nu}(3) symmetry.The calculated
\rho^2(E0, 0^+_2\rightarrow0^+_1)\times10^3 value also exactly reproduced the observed data. Both the theoretical and experimental properties ofE0 transition indicate that the 0^+_2 state in 124Te may be interpreted as the lowest state of the first-excited family of intrinsic levels at the critical point of second-order\rm{U}_{\pi \nu}(5) -\rm{O}_{\pi \nu}(6) transition. Therefore, we conclude that the 124Te is a possible nucleus at the critical point of the second-order phase transition from vibration to unstable rotation, and such a critical point exhibits a slight triaxial rotation. The 0^+_2 state in 124Te can be interpreted as the lowest state of the first-excited family of intrinsic levels in the critical point symmetry. However, it should be mentioned that some of the measuredB(E2) andB(M1) values have a large uncertainty and the experimental data ofB(M1, 1^+_1\rightarrow 0^+_2) is still missing. More theoretical and experimental investigations should focus on these aspects.We are grateful to Prof. CaiWan Shen, and Dr. XiaoBao Wang of Huzhou University, Prof. YuXin Liu of Peking University, and Prof. GuiLu Long of Tsinghua University for their helpful discussions and good suggestions.
Description of the critical point symmetry in 124Te by IBM-2
- Received Date: 2018-10-27
- Available Online: 2019-02-01
Abstract: Based on the neutron and proton degrees of freedom, low-lying energy levels,