Description of the critical point symmetry in 124Te by IBM-2

TL;DR

This paper uses the neutron-proton interacting boson model to analyze the energy levels and transition strengths of 124Te, suggesting it is at a critical point of a second order phase transition with triaxial rotation.

Contribution

It demonstrates that 124Te exhibits properties of a critical point symmetry at a phase transition, supported by calculations consistent with experimental data.

Findings

124Te is at the critical point of a second order phase transition.

The 0+2 state is interpreted as the lowest intrinsic level in the critical point symmetry.

Calculated transition strengths agree well with experimental data.

Abstract

Based on the neutron and proton degrees of freedom, the low-lying energy levels, the $E2$, $M1$ and $E0$ transition strengths of nucleus $^{124}$Te have been calculated by the neutron-proton interacting boson model. The calculated results are quite consistent with the experimental data. By comparing the key observables of states at the critical point of $\textrm{U}_{\pi \nu}(5)$--$\textrm{O}_{\pi \nu}(6)$ transition with experimental data and calculated results, we show that the $^{124}$Te is a possible nucleus at the critical point of the second order phase transition from vibration to unstable rotation and such a critical point contains somewhat triaxial rotation. The 0$^+_2$ state of $^{124}$Te can be interpreted as the lowest state of the first excited family of intrinsic levels in the critical point symmetry.

Tables

	Table 1(continued)
$J_i^{\pi }\to$$J_f^{\pi }$	Expt.	IBM-2
$2_6^{+}$$\to$$2_2^{+}$	1.6${}_5{}^{}{}_{}{}^5$	1.67
$2_6^{+}$$\to$$4_1^{+}$	1.7${}_4{}^{}{}_{}{}^5$	0.00
$2_6^{+}$$\to$$2_1^{+}$	0.046${}_6{}^{}{}_{}{}^8$	0.23
$2_6^{+}$$\to$$0_1^{+}$	0.12${}_2{}^{}{}_{}{}^3$	0.01
$4_4^{+}$$\to$$4_1^{+}$	2.6${}_7{}^{}{}_{}{}^9$	0.10
$4_4^{+}$$\to$$2_1^{+}$	0.051${}_{22}{}^{}{}_{}{}^{28}$	0.72
$2_7^{+}$$\to$$2_1^{+}$	4.0${}_4{}^{}{}_{}{}^4$	0.56
$2_7^{+}$$\to$$2_2^{+}$	0.20${}_{11}{}^{}{}_{}{}^{10}$	0.00
$2_7^{+}$$\to$$4_1^{+}$		1.10

Table 2. Table 2: Comparison of the key observables of the states at the critical point of the U π ​ ν ​ ( 5 ) subscript U 𝜋 𝜈 5 \textrm{U}_{\pi\nu}(5) – O π ​ ν ​ ( 6 ) subscript O 𝜋 𝜈 6 \textrm{O}_{\pi\nu}(6) transition (taken from Ref. Iachello00 and denoted as CPST), the experimental data of 124 Te (taken from Refs. Garrett18 ; Hicks17 ; Rikovska89 ; Hicks08 ; Doll00 and labeled as 124 Te) and the calculated results (labeled as IBM-2).

	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 3. Table 3: Comparison of the calculated M ​ 1 𝑀 1 M1 transition strengths (in units of μ N 2 subscript superscript 𝜇 2 𝑁 \mu^{2}_{N} ) and the experimental data for 124 Te. The experimental data are taken from Refs. Hicks08 ; Doll00 ; Hicks12 ; Guliyev02 .

Transition	Expt.	IBM-2
$2_2^{+}$$\to$$2_1^{+}$	0.0003	0.006
$2_3^{+}$$\to$$2_2^{+}$	0.004	0.025
$2_3^{+}$$\to$$2_1^{+}$	0.013	0.072
$2_4^{+}$$\to$$2_2^{+}$	0.001	0.031
$2_4^{+}$$\to$$2_1^{+}$	0.004	0.075
$1_1^{+}$$\to$$0_1^{+}$	0.04	0.029
$1_1^{+}$$\to$$0_2^{+}$		0.034

Equations

$$\hat{H}={\epsilon }_{d\pi }{\hat{n}}_{d\pi }+{\epsilon }_{d\nu }{\hat{n}}_{d\nu }+{\kappa }_{\pi \nu }{\hat{Q}}_{\pi }\cdot {\hat{Q}}_{\nu }.$$

$$\hat{H}={\epsilon }_{d\pi }{\hat{n}}_{d\pi }+{\epsilon }_{d\nu }{\hat{n}}_{d\nu }+{\kappa }_{\pi \nu }{\hat{Q}}_{\pi }\cdot {\hat{Q}}_{\nu }+{\omega }_{\pi \nu }{\hat{L}}_{\pi }\cdot {\hat{L}}_{\nu }+{\hat{M}}_{\pi \nu },$$

$$B(E2,J^{\prime }\to J)=\frac{1}{2J^{\prime }+1}|J⟨\Vert \hat{T}(E2)\Vert J^{\prime }⟩{|}^2.$$

$$\hat{T}{(M1)}=\sqrt{\frac{3}{4\pi}}(g_{\nu}\hat{L}_{\nu}+g_{\pi}\hat{L}_{\pi}),\$$

$$\rho (E0,J^{\prime }\to J)=\frac{Z}{e{R}^2}[{\beta }_{0\pi }⟨J\Vert {\hat{T}}_{\pi }(E0)\Vert J^{\prime }⟩+{\beta }_{0\nu }⟨J\Vert {\hat{T}}_{\nu }(E0)\Vert J^{\prime }⟩],$$

Full text

Key words: 124Te, low-lying structure, the critical point, $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition.

I Introduction

How to characterize the underlying structure of 124Te is a difficult task and has attracted considerable interests in variety of theoretical and experimental studies Garrett18 ; He17 ; Hicks17 ; Garrett16 ; Sabria16 ; Saxena14 . The 124Te lies outside the major shell $Z=50$ with two protons, and its ratio $E(4^{+}_{1}/2^{+}_{1})=2.07$ 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 more $\gamma$-unstable or O$(6)$ behavior, but the $E2$ decays of 0${}^{+}_{2}$ state do not support the $\gamma$-soft rotational nature.

On the other hand, shape coexistence in the Te isotopes is still elusive Garrett16 . The appearance of strong $\rho(E0,0^{+}_{2}\rightarrow 0^{+}_{1})$ transition is an important spectroscopic fingerprint that mostly supports the shape coexistence in nucleus Heyde11 . For 124Te, the experimental value of the electric monopole transition strength $\rho^{2}(E0,0^{+}_{2}\rightarrow 0^{+}_{1})\times 10^{3}$ is $12\pm 3$ T05 . This value is very similar to the corresponding transition in the Cd isotopes, the latter demonstrates the firm evidence for the nature of shape coexistence, leading to strong support for deformed intruder structure of the first excited 0+ state in 124Te isotopes, but not providing conclusive proof for shape coexistence in the 124Te yet Garrett16 . Meanwhile, the systematic research on the known experimental information suggested that 124Te may be an example of nucleus at the $E(5)$ critical point Clark04 . The recent investigation indicates that 124Te exhibits $E(5)$-like structure Sabria16 ; Ghit08 . But there are some debates on understanding the low-lying structure of the 124Te ( see, e.g., Refs.Hicks17 ; Ghit08 ; Guliyev02 and references therein).

The $E(5)$ and $X(5)$ symmetries originally developed by Iachello Iachello00 ; Iachello01 correspond to an exact solution of the Bohr Hamiltonian for $\gamma$-independent potentials and an approximate solution of the Bohr Hamiltonian for $\gamma\approx 0^{\circ}$, respectively. From the Bohr Hamiltonian with $\gamma$ frozen at $\gamma=30^{\circ}$, a four dimensional critical point symmetry model called $Z(4)$ has been introduced Bonatsos2005 . Most recently, another four dimensional critical point symmetry called $T(4)$ is obtained from the Bohr Hamiltonian with the $\beta$-soft potential and for a fixed value of $\gamma$ with $0^{\circ}\leq\gamma\leq 30^{\circ}$ Zhangyu2017 .

In nuclear system, $E(5)$ symmetry can be used to describe a nucleus at the critical point of the second order phase transition from a spherical vibrator to a $\gamma$-soft rotor. In the interacting boson model (IBM-1) Iachello87 , which do not distinguish the proton boson and neutron boson, the $E(5)$ represents a nucleus that is located at the critical point of the shape phase transition between the U$(5)$ and O$(6)$ symmetries. Since the neutron and proton degrees of freedom are explicitly taken into account, the proton-neutron interacting boson model (IBM-2)Iachello87 possesses a complex phase diagram Caprio05 ; Cejnar09 . The critical point of second order phase transition can occur at the phase transition from $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ symmetry, and can be subsumed to the critical point of the second order phase transition in IBM-1 Caprio05 ; Cejnar09 ; Cejnar10 . Compared with their neighboring isotopes, nuclei at or close to the critical point of shape phase transition demonstrate dramatic changes in the properties of the low-lying states, such as the energy levels, the $E2$ transition strengths and $E2$ branching ratios Naz18 . But there is no discussion on the critical point symmetry in the real nuclei for the phase transition between $\textrm{U}_{\pi\nu}(5)$ and $\textrm{O}_{\pi\nu}(6)$ in the IBM-2 phase diagram, so far.

The low-lying levels of 124Te have recently been examined with the (${n,n^{\prime},\gamma}$) reaction. The spins, level energies, $B(E2)$ transition probabilities, and multipole-mixing ratios were obtained Hicks17 . The experiment provides us an opportunity to deeply understand 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 special attention paid to the key observables of the critical point of the second order transition between $\textrm{U}_{\pi\nu}(5)$ and $\textrm{O}_{\pi\nu}(6)$. We also calculate the low-lying energy levels, the $E2$, $M1$ and $E0$ transition strengths and compare the predications of the critical point symmetry of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ 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 Sec. II, 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 predications of the critical point symmetry are presented in Sec. III. Finally, out concluding remarks and summary are stated in Sec. IV.

II Theoretical framework

The IBM-2 is an algebraic model, in which valence nucleons are coupled to form $s_{\rho}$ bosons (angular momentum $L$=0) and $d_{\rho}$ bosons (angular momentum $L=2$) with $\rho=\pi$ standing for proton bosons and $\rho=\nu$ standing for neutron bosons, respectively. The microscopic structure of the model suggests that only two terms are very important. One is the pairing interaction between identical nucleons, the other is the quadrupole-quadrupole interaction between non-identical nucleons. The simple standard IBM-2 Hamiltonian Iachello87 ; Otsuka92 is written as

[TABLE]

where $\hat{n}_{d\rho}=d^{{\dagger}}_{\rho}\cdot\tilde{d_{\rho}}\,$ and $\hat{Q}_{\rho}=(s^{{\dagger}}_{\rho}\tilde{d}_{\rho}+d^{{\dagger}}_{\rho}{s}_{\rho})^{(2)}+\chi_{\rho}(d^{{\dagger}}_{\rho}\tilde{d}_{\rho})^{(2)}\,$ represent $d$-boson number operator and quadrupole operator, respectively. The parameter ${\chi_{\rho}}$ determines the type of the deformation of the quadrupole operator. $\varepsilon_{d\rho}$ is the energy of the d bosons relative to the $s$ bosons, $\kappa_{\pi\nu}$ is the strength of the quadrupole-quadrupole interaction between neutron boson and proton boson.

The Hamiltonian of Eq.(1) has a much richer shape phase structure, which contains $\textrm{U}_{\pi\nu}(5)$, $\textrm{O}_{\pi\nu}(6)$, $\textrm{SU}_{\pi\nu}(3)$, $\overline{\textrm{SU}_{\pi\nu}(3)}$, and $\textrm{SU}^{\ast}_{\pi\nu}(3)$ dynamical symmetries, corresponding to the spherical vibrator, $\gamma$-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 in between the different dynamical symmetries in the IBM Caprio05 ; Cejnar09 ; Zhangyu20141 . Although the standard IBM-2 Hamiltonian can give us more clear space of dynamical symmetry, one has to add other physical dominant interactions in order to described real nuclei more accurately. We use the following IBM-2 Hamiltonian in this paper Iachello87 ; Pietralla08 ; Zhang15 ,

[TABLE]

where $\hat{L_{\rho}}=\sqrt{10}[d^{{\dagger}}_{\rho}\cdot\tilde{d_{\rho}}]^{(1)}$ is the angular momentum operator with a dipole proton-neutron interaction parameter $\omega_{\pi\nu}$, and $\hat{M}_{\pi\nu}=\lambda_{2}(s^{{\dagger}}_{\pi}d^{{\dagger}}_{\nu}-s^{{\dagger}}_{\nu}d^{{\dagger}}_{\pi})^{(2)}\cdot(s_{\pi}\tilde{d}_{\nu}-s_{\nu}\tilde{d}_{\pi})^{(2)}$ $+\sum_{k=1,3}\lambda_{k}(d^{{\dagger}}_{\pi}d^{{\dagger}}_{\nu})^{(k)}\cdot(\tilde{d}_{\pi}\tilde{d}_{\nu})^{(k)}\,$ is the Majorana interaction, the Majorana parameters $\lambda_{k}$ (k=1,2,3) represent the strength of the Majorana interaction. If one does not include the Majorana term in Eq. (2), the adopted parameters in the remaining four terms of Eq. (2) indicate that the nucleus described by the IBM-2 may locate on the the plane of $\textrm{U}_{\pi\nu}(5)$-$\textrm{O}_{\pi\nu}(6)$-$\textrm{SU}^{\ast}_{\pi\nu}(3)$, because the $\hat{L}_{\pi}\cdot\hat{L}_{\nu}$ only generate the physical rotational group $\textrm{SO}_{\pi\nu}(3)$ of the IBM-2 Iachello87 ; Caprio05 ; Cejnar09 . On the other hand, in Ref.zhangyu20142 the authors proposed a new algebraic model $F(5)$ based on the Euclidean dynamical symmetry in five dimensions ($\rm{Eu(5)}$), which can build a symmetry intermediate between the $E(5)$ and $X(5)$ symmetries Zhangyu20141 . However, the $F(5)$ cannot directly be defined in the IBM-1 or IBM-2 due to the non-compactness of the $\rm{Eu(5)}$ group zhangyu20142 .

The decay properties of the low-lying states, such as the $B(E2)$ transition probabilities, the magnetic dipole $B(M1)$ transition strengths and the $\rho^{2}(E0)$ values between lowest 0+ states are the signatures of phase structure. In the IBM-2, the E2 transition strength is given by the following expression

[TABLE]

where $J^{\prime}$ and $J$ 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 $E$2 operator, where the operator $Q_{\rho}$ is the same as in Eq.(1). The parameters $e_{\pi}$ and $e_{\nu}$ are the effective charges of proton bosons and neutron bosons, respectively. The values of $e_{\nu}$ and $e_{\pi}$ could be taken differently.

In the proton-neutron interacting boson model, the magnetic dipole $M1$ transition operator is defined as

[TABLE]

where the $\hat{L}_{\rho}$ is the same as in Eq.(2). The $g_{\pi}$ and $g_{\nu}$ are the effective proton and neutron boson $g$-factors, repectively. Typically, one can take the values of $g_{\pi}=1$, $g_{\nu}=0$ in the calculations Otsuka92 .

The $E0$ transition matrix element $\rho$ in the IBM-2 is written as Giannatiempo93 ; Zhang18

[TABLE]

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 of $e$fm2, respectively. The $E$0 operator is expressed as $\hat{T}{(E0)}=\beta_{0\pi}\hat{T}_{\pi}{(E0)}+\beta_{0\nu}\hat{T}_{\nu}{(E0)}=\beta_{0\pi}\hat{n}_{d\pi}+\beta_{0\nu}\hat{n}_{d\nu}$, where the $\hat{n}_{d\rho}$ is the same as in Eq.(1).

III Results and discussion

The 124Te situates at the position between vibrational nucleus and deformed nucleus Vanhoy04 . The IBM provides a powerful tool to describe the nuclear shapes and the shape phase transitions. With this approach, a lot of works have been devoted to describe the properties of the low-lying states of 124Te isotope. In IBM-1, the early work regarded the 124Te as an example of O$(6)$ symmetry, but the energy levels of yrast states of 124Te differ from the O(6) limit Robinson83 . The further systematic investigations found that 124Te was close to the U$(5)$ symmetry Kern95 ; Pascu10 . Moreover, recent results of experiments and calculations indicated that 124Te may posses the $E(5)$ features Hicks17 ; Clark04 ; Sabria16 ; Ghit08 . In IBM-2, the calculations with and without mixed configuration suggested that 124Te is an O$(6)$-like nucleus, but the $B(E2)$ decay pattern is not consistent with this symmetry RIKOVSKA89 . The subsequent study shows that neither O$(6)$ nor U$(5)$ symmetries can describe the 124Te well Warr98 , although the potential-energy surface of 124Te exhibits an obvious manifestation of O$(6)$ limit SubberJf87 . Another recent experiment and IBM calculation support that 124Te shows the soft triaxial behavior Saxena14 . Although the debate on the low-lying structure of the 124Te still exists, it is indubitable that all these investigations indicated that 124Te is a transitional nucleus between vibration U$(5)$ symmetry and $\gamma$-unstable O$(6)$ symmetry in IBM, and might accompany with somewhat soft triaxial rotation.

To describe the low-lying structure of 124Te, we select the doubly closed shell ${}^{132}_{50}$Sn82 as the inert core. There are $N_{\pi}=1$ particle-like boson beyond the $Z=50$ major shell and $N_{\nu}=5$ hole-like neutron bosons from the $N=82$ shell closure, and the total number of bosons is $N_{B}=6$. Considering that the proton boson is particle-like and the neutron bosons are hole-like based on the different single particle orbitals in 124Te, we take different relative energies for $d$ neutron and proton boson in this case, $i.e.$, $\varepsilon_{d\pi}\neq\varepsilon_{d\nu}$ just as in Refs.Dejbakhsh92 ; zhangding14 ; zhangmulett16 . To describe the different type of valence neutrons and protons exhibiting opposite intrinsic quadrupole deformation, and obtain the $\gamma$-unstable O$(6)$ symmetry, we adopt the $\chi_{\pi}+\chi_{\nu}=0$. In order to introduce the somewhat character of 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, but it does not affect the wave vectors Pietralla98 . 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 experimental data especially for the 2${}^{+}_{1}$ state, we obtain the parameters of IBM-2 in this work 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 Otsuka85 . In Figure 1, we compare the calculated level energies with the experimental results Hicks17 ; Egidy06 . The experimental level energies in Figure 1 are separated into different bands based only on their order of appearance, where only the low-spin positive parity states with uniquely assigned spin and parity are displayed.

Figure 1 shows that the theoretical levels are good consistent with the corresponding experimental ones. 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 122-130 Te isotopes Hicks17 . By introducing the dipole interaction term into the Hamiltonian Nomura11 ; Nomura16 ; Zhangmu17 , the structure of the yrast bands including the 6${}^{+}_{1}$ level are reproduced by the theoretical predications. Similar to the 6${}^{+}_{1}$ state, the observed energy of 4${}^{+}_{2}$ level is only changed about 135 keV from 122Te to 130Te isotopes. The present calculation reproduces the experimental energy of the 4${}^{+}_{2}$ level nicely. 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 obviously 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 predications for higher energy states, such as 2${}^{+}_{6}$, and 2${}^{+}_{7}$ states, which is a general feature of this model Kim96 . For the 0${}^{+}_{4}$ state, the calculated level is significantly higher than the experimental one. The IBM-2 with the mixed configuration shows that 0${}^{+}_{4}$ is an intruder state in 124Te and locates outside the IBM-2 space RIKOVSKA89 .

In seeking to investigate the properties of the $E2$ transitions in 124Te, the effective charges of proton and neutron bosons were determined to reproduce the experimental $B(E2)$ values. Using the same manner as in Refs.Elhami07 ; zhangmu16 and exactly fitting to the experimental data of $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})=31(5)$ W.u., we obtain $e_{\pi}=3.780$ $\sqrt{\rm{W.u.}}$, $e_{\nu}=1.670$ $\sqrt{\rm{W.u.}}$. The theoretical calculation of $B(E2)$ values for 124Te in comparison with the available experimental data Garrett18 ; Hicks17 ; Ghit08 ; Rikovska89 ; Hicks08 ; Doll00 is given in Table 1.

Table 1 shows that the computed $\emph{B}(E2)$ transition strengths are in overall agreement with the experimental data, although most cases of the experimental value have a large uncertainty. In more detail, the typical strongly collective $E2$ transitions with tens of Weisskopf units are described by the theoretical predications very well, some of them consist with each other within the experimental uncertainty. It is noteworthy that the experimental $\emph{B}(E2,4^{+}_{1}\rightarrow 2^{+}_{1})$ transition strength is $35.9(\pm 17)$ W.u. in Ref.Hicks17 , which is taken from Ref.Saxena14 . In Ref.Doll00 , 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.Rikovska89 , we adopt the experimental averaged value 54.51 W.u. for this transition, the detail can be seen in the Ref.Mardirosian1984 , one can find that the experimental value agrees with the present calculated value. At the same time, the experimental relatively strong $B(E2)$ transitions, which can be comparable to 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 $\emph{B}(E2,4^{+}_{3}\rightarrow 2^{+}_{2})$ transition strength. Furthermore, the calculated results are in good description of the properties of the experimental weakly collective $E2$ transitions with about one or even less than one W.u.. The theoretical $E2$ 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 a little higher than the experimental data.

To identify where the 124Te can be placed in the $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ transition, we focus on a set of key observables Casten00 , such as the energy ratios $R_{4_{1}/2_{1}}=E(4^{+}_{1})/E(2^{+}_{1})$, $R_{2_{2}/2_{1}}=E(2^{+}_{2})/E(2^{+}_{1})$, $R_{0_{2}/2_{1}}=E(0^{+}_{2})/E(2^{+}_{1})$, $R_{0_{3}/0_{2}}=E(0^{+}_{3})/E(0^{+}_{2})$, and the $B(E2)$ ratios $R_{B,42}=B(E2,4^{+}_{1}\rightarrow 2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$, $R_{B,22}=B(E2,2^{+}_{2}\rightarrow 2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ and $R_{B,02}=B(E2,0^{+}_{2}\rightarrow 2^{+}_{1})/B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$, which reflect the characteristics of the nucleus behavior at critical point of the phase transition from $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ in IBM-2 space Caprio05 , and are the most crucial nuclear structure indicators Casten88 . Some of the indicators can even distinguish the first order quantum phase transition from the second order quantum phase transition IZ04 ; Liu07 ; Bonatsos081 . The values of these key observables at the critical point of the phase transition from $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ for the infinite numbers of bosons Iachello00 , as well as the experimental data and the calculated results in 124Te are listed in Table 2 for comparison.

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 the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. The experimental ratio $R_{0_{3}/0_{2}}$ is very close to the value of CPST. Meanwhile, the experimental energy of the $0^{+}_{2}$ state relative to $E(2^{+}_{1})$ satisfactorily matches the predicated value of CPST. The experimental $R_{4_{1}/2_{1}}$ lies between U${}_{\pi\nu}(5)$ ($R_{2_{2}/2_{1}}=2.00$) and $\textrm{O}_{\pi\nu}(6)$ ($R_{2_{2}/2_{1}}=2.50$) limits, although it deviates slightly from the predicated one. On the $B(E2)$ ratios, the experimental $R_{B,42}$ is well reproduced by the theoretical predication and clearly points to a structure intermediate between U${}_{\pi\nu}(5)$ ($R_{B,42}=2.00$) and $\textrm{O}_{\pi\nu}(6)$($R_{B,42}=10/7$). Meanwhile, the experimental value of $R_{B,22}$ approaches the predicted value of the critical point theory, too. In particular, the information of $R_{B,02}$ is a signature for identifying the $\textrm{U}_{\pi\nu}(5)$ from the $\textrm{O}_{\pi\nu}(6)$ symmetry. In $\textrm{U}_{\pi\nu}(5)$ limit the $B(E2,0^{+}_{2}\rightarrow 2^{+}_{1})$ transition strength is 2 times larger than $(B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$. In $\textrm{O}_{\pi\nu}(6)$ limit the $B(E2,0^{+}_{2}\rightarrow 2^{+}_{1})$ transition is forbidden Stachel82 . Here, both the experimental and theoretical values of the $R_{B,02}$ can correctly reflect the nature of the $B(E2,0^{+}_{2}\rightarrow 2^{+}_{1})$ transition, although the calculated value overestimates about 1.5 times than experimental one. Therefore, all the available experimental information on the key observables for 124Te is in good agreement with the predictions of critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. Meanwhile, Table 2 also shows that the present calculation of the characteristic feature of 124Te is remarkable. However, the $R_{0_{2}/2_{1}}$ of CPST seems larger than the IBM’s in Table 2, the reason is that the calculation of CPST is given for the infinite$-N_{B}$ limit Iachello00 ; Casten00 . For the finite number of bosons, Ref.Caprio05 gives that the $R_{0_{2}/2_{1}}$ of a nucleus at the critical point of the phase transition from $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ with $N_{\pi}=N_{\nu}=5$ is 2.48, which agrees the result of present calculation. From the above discussion, we conclude that the 124Te may be a nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. Consequently, the $0^{+}_{2}$ state is the possible lowest 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 Pietralla08 . In IBM-2, the pure $\textrm{O}_{\pi\nu}(6)$ symmetry can be obtained only with $\chi_{\pi}=\chi_{\nu}=0$. In this case, there does not exist any asymmetry between $\chi_{\pi}$ and $\chi_{\nu}$. It means that the mixed symmetry components could not mix into the low-lying states. Therefore, the $M1$ transitions between these states are forbidden in $\textrm{O}_{\pi\nu}(6)$ limit Otsuka92 . The theoretical predictions show that the $B(M1)$ transition strengths among the low-lying states almost vanish, when a nucleus is at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition Caprio05 . For the scissors mode state 1${}^{+}_{1}$, the $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ transition strength is $3N_{\pi}N_{\nu}(g_{\nu}-g_{\pi})^{2}/{4\pi(2N_{B}+1)}$ for $\textrm{O}_{\pi\nu}(6)$ limit, but it is forbidden for $\textrm{U}_{\pi\nu}(5)$ limit. The $B(M1,1^{+}_{1}\rightarrow 0^{+}_{2})$ value is $3N_{\pi}N_{\nu}(g_{\nu}-g_{\pi})^{2}/{\pi N_{B}(N_{B}-1)}$ for $\textrm{U}_{\pi\nu}(5)$ limit, whereas vanish for $\textrm{O}_{\pi\nu}(6)$ limit Pietralla08 ; Backer86 ; Sevrin87 . At the critical point of $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition, the 1${}^{+}_{1}$ state has an allowed $M1$ decays to first and second $0^{+}$ states. In the case of 124Te, the $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ is 0.10 $\mu^{2}_{N}$ for $\textrm{O}_{\pi\nu}(6)$ limit, and $B(M1,1^{+}_{1}\rightarrow 0^{+}_{2})$ is 0.50 $\mu^{2}_{N}$ for $\textrm{U}_{\pi\nu}(5)$ limit with the typical values $g_{\pi}=1$, $g_{\nu}=0$. Indeed, only a few $M1$ transitions with very small absolute values of $B(M1)$ among the low-lying states in 124Te have been measured Hicks08 ; Georgii95 . By using the Eq.(4) and also taking $g_{\pi}=1$ and $g_{\nu}=0$, the calculated $M1$ transition probabilities, as well as the corresponding available experimental data for the scissors mode and the low-lying states in 124Te are given in Table 3.

As can be seen from Table 3, all the experimental $B(M1)$ transition strengths between the lowest 2+ states are of the order of about 0.01 $\mu^{2}_{N}$ or even less, which is far from the typical $B(M1)$ value of the mixed symmetry state. This demonstrates that the observed $M1$ transitions are in qualitative agreement with the predictions of the critical point symmetry of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. At the same time, Table 3 shows that the calculated results reproduce the characteristics of the $M1$ transitions among low-lying states very fairly. In particular, the calculated $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ strength is close to the experimental one, which is comparable the predicted value of the critical point between the $\textrm{U}_{\pi\nu}(5)$ and $\textrm{O}_{\pi\nu}(6)$ limits. This indicates that both the experimental and the calculated results of the $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ transition reflect the property of the nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. Unfortunately, the $B(M1,1^{+}_{1}\rightarrow 0^{+}_{2})$ strength in 124Te has not been measured, the calculated value for this transition is 0.035 $\mu^{2}_{N}$, locating between the $\textrm{U}_{\pi\nu}(5)$ and $\textrm{O}_{\pi\nu}(6)$ transition similar to the case of $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ transition. All these results indicate that both the calculated and experimental $M1$ transitions are consistent with the characteristic behavior of the 124Te nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition.

However, some deviations between the theoretical and experimental $M1$ transition strengths also are displayed in Table 3, all the calculated $M1$ values are a little larger than the corresponding experimental data. In IBM-2, $\chi_{\pi}=-\chi_{\nu}\neq 0$ leads to spectrum having many of the 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 Otsuka92 ; zhangmusci18 . On the other hand, $\chi_{\pi}=-\chi_{\nu}\neq 0$ makes the nucleus have some features of the $\textrm{SU}^{\ast}_{\pi\nu}(3)$ symmetry, because the proton and neutron have opposite types of deformation Cejnar09 . The calculated $B(M1)$ values among the lowest states are systematically slightly larger than the experimental data, which implies that the description of 124Te with the character of the $\textrm{SU}^{\ast}_{\pi\nu}(3)$ is slightly rather than the actual nucleus. Furthermore, the small $B(M1)$ values among the low-lying states in 124Te also indicate that 124Te may contain some components of $\textrm{SU}^{\ast}_{\pi\nu}(3)$ symmetry. Therefore, the 124Te may be the nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition and with somewhat $\textrm{SU}^{\ast}_{\pi\nu}(3)$ symmetry.

The electric monopole transition strengths between 0+ states can be considered as a signature of shape coexistence Heyde11 , quantum phase transitions Nomura17 , and can reflect the properties of $E(5)$ dynamical symmetry Clark04 . In fact, the $\rho^{2}(E0,0^{+}_{2}\rightarrow 0^{+}_{1})$ value of 124Te has been measured T05 . But the $\rho^{2}(E0)$ value on the first excited 0+ state in 124Te is a puzzle for a long time as mentioned in the introduction Garrett16 . In IBM, the $E0$ transitions do not occur in the U(5) dynamical limit, because $E0$ operator is proportional to $\hat{n}_{d}$ Leviatan16 . While in O(6) limit, the selection rules require the $E0$ transition from the 0${}^{+}_{2}$ to ground state is forbidden Wood99 . For the predicted $E(5)$ symmetry, the 0${}^{+}_{2}$ level, which is the lowest member (zero phonon) of the first excited family, should have an allowed $E0$ branch to the 0${}^{+}_{1}$ state. However, so far the studies of the behavior of $E(5)$ symmetry in 124Te have not provide a detailed analysis of the properties of $E0$ 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-1Caprio05 ; Pietralla08 . The critical point symmetry of $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ bears considerably resemblance to the predictions of the ${E}(5)$ symmetry Caprio05 . Subsequently, the $0^{+}_{2}$ level should also have an allowed $E0$ 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.Giannatiempo91 , namely, $\beta_{0\nu}=0.25$ and $\beta_{0\pi}=0.10$ $e$fm2. The calculated $\rho^{2}(E0,0^{+}_{2}\rightarrow 0^{+}_{1})\times 10^{3}$ value is $11.00$, which is consistent with the experimental data within the experimental uncertainty. The theoretical and experimental $E0$ 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 $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition.

It is well known that the low-lying yrast states in other Te isotopes such as 112,114Te show the similar behavior, i.e., typical vibrational like spectra but the other properties deviate from vibrator Saxena14 ; Moller2005 ; Qi2016 ; Doncel2017 . All these Te isotopes locate 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 point of IBM model, 112Te is below the midshell 118Te, it has $N_{\nu}=5$ bosons beyond $N=50$ major shell. Both 112Te and 124Te have the same numbers of bosons with $N_{\pi}=1$ and $N_{\nu}=5$, therefore, they show the similar properties of spectrum. As for 114Te, the measured ratio of $B(E2,4^{+}_{1}\rightarrow 2^{+}_{1})$ to $B(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ is smaller than one, which is very unusual deformation, no theoretical models can give a satisfied description Doncel2017 .

IV Conclusion

In summary, the low-lying structure of the 124Te has been investigated within the framework of IBM-2. The calculated low-lying energy levels are good consistent with the corresponding experimental data. In particular, the 6${}^{+}_{1}$, 4${}^{+}_{2}$ levels, which are remaining at a nearly constant excitation energy from 122Te to 130Te isotopes, have been reproduced by calculated results very satisfactorily. Meanwhile, the description of the first two excitated 0+ states in energy sequence is very well, although the 0${}^{+}_{2}$ level is obviously 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 $\emph{B}(E2)$ transition strengths are in the overall agreement with the experimental data. The observed typical strongly collective $E2$ transitions with tens of Weisskopf units, and the most of the experimental relatively strong $B(E2)$ transitions comparable to the experimental $\emph{B}(E2,2^{+}_{1}\rightarrow 0^{+}_{1})$ transition probability with dozens of W.u., are reproduced by the theoretical predications nicely. Furthermore, the calculated results provide the perfect description of the properties of the experimentally weakly collective $E2$ transition with about one or even less than one W.u.. Particularly, the theoretical $E2$ 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${}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition with the experimental data and 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 $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ very well. All these information on the key observables indicates that the 124Te may be a nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition. Consequently, the $0^{+}_{2}$ state is the possible lowest state of the first excited family of 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 predications of the critical point of the transition from $\textrm{U}_{\pi\nu}(5)$ to $\textrm{O}_{\pi\nu}(6)$ well, and the overall agreement between the experiment and the theory for characteristics of the $B(M1)$ transition strengths is perfect. Especially, both the experimental and theoretical $B(M1,1^{+}_{1}\rightarrow 0^{+}_{1})$ values are located between the $\textrm{U}_{\pi\nu}(5)$ and $\textrm{O}_{\pi\nu}(6)$ limits. By analyzing the calculated results and the structure parameters of quadrupole operator in this work, all these quantities support that the 124Te may be the nucleus at the critical point of the $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ transition and with somewhat feature of $\textrm{SU}^{\ast}_{\pi\nu}(3)$ symmetry.

The calculated $\rho^{2}(E0,0^{+}_{2}\rightarrow 0^{+}_{1})\times 10^{3}$ value also reproduced exactly the observed data. Both the theoretical and experimental properties of $E0$ 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 $\textrm{U}_{\pi\nu}(5)$–$\textrm{O}_{\pi\nu}(6)$ 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 contains somewhat 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 a few of the measured $B(E2)$ and $B(M1)$ values have a large uncertainty and the experimental data of $B(M1,1^{+}_{1}\rightarrow 0^{+}_{2})$ is still missing. More investigations on the experiment and the theory are needed to focus on these aspects.

Acknowledgements.

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. This work is supported by the National Natural Science Foundation of China under grant numbers 11475062, 11747312 and 11147148.