1. Introduction

The Cd isotopes are of specific interest since they have only two fewer protons than the individual closed-shell Sn nuclei. The large abundance of stable isotopes in combination with the interesting properties of the closed proton shell near Z=50 and the presence of neutrons in the middle of the N=50-82 shell make the Z=50 region favored for nuclear structure studies ^[1-4]. Schraff-Goldhaber and Wesener observed that The Cd isotopes have low lying states that reflect the quadrupole-vibrational excitations of a spherical-equilibrium surface ^[5]. lachello and Arima ^[6] successfully described the collective nuclear properties in intermediate- mass nuclei using the Interacting Boson Model (IBM-1). Moreover, the IBM-1 model generated the U(6) group algebra, that then produces three subgroup symmetries: U(5), SU(3), and O(6). These three dynamics are associated with a vibrational, rotational, and 7-soft nuclei, respectively ^[2,3]. However, many researchers have suggested that nuclei may have an intermediate structure consisting of the transitions U(5)-SU(3), U(5)-O(6), and SU(3)-O(6) ^[4,5].

Great deal of research on the structure of electromagnetic transitions and energy levels in even-even Cd isotopes have been carried out ^[6-12]. The stable isotopes of ^110,112Cd have already been explained as collective nuclei with multiphonon excitations. Morrison and Smith ^[13] presented four distinct constant boson interactions in order to illustrate the framework of ^108,116Cd isotopes for states with angular momentum (L=1 to 6). Pignanelli et al. ^[14] used the interacting boson model to study the octupole excitations in nuclei with masses between 98 and 150. The low-spin state in even-even ^106-112,116Cd isotopes was examined utilizing γ-ray (in/off-beam) and conversion-electron spectroscopy ^[15]. Long et al. ^[7] used IBM-1 to explain the low-lying levels and high-spin states of ^116,118,120 Cd. The same procedure was also carried out by Bollaert et al. ^[16]. Moreover, Kadi et al. ^[17] used the (n,n', γ) reaction to investigate the lifetimes of various levels as well as the decay parameters of multiphonon quadrupole vibrational states and invasion structures in ¹¹⁶Cd. They found that the intruder pattern is fully confirmed in the ¹¹⁶Cd nuclei. likewise, Gade et al. ^[18] evaluated the electric and magnetic dipole excitations of ¹⁰⁸Cd nuclei and estimated the lifetimes of eight dipoles. Furthermore, the researchers demonstrated a transition path between the dynamic U(5) and O(6) limits for the ¹⁰⁸Cd using the IBM-2 framework. Garrett et al. ^[19] used the interaction boson model to analyze the systematic variation of ^110-114Cd isotopes. Their calculations show that Cd isotopes at the two-photon levels reflect the data quite well, while the calculations for the 0⁺ states in ¹¹⁶Cd fail completely. Systematic variations occur across the Cd isotopic chain at the three-phonon level, demonstrating the breakdown of vibrational motion in low-spin states. Hossian et al. ^[20] used IBM-1 to study the energy level of ^104-122Cd nuclides. They noted that Cd isotopes have a vibrational symmetry U(5). The resulting Hamiltonian was utilized by Nomura etal. ^[21] to calculate the low-lying excitation bands, quadrupole and monopole electrical transition rates for the even-even ^108-116Cd. Many intruder states were predicted, which corresponded to empirical data. Besides, Leviatan et al. ^[22] show that vibrational symmetry is retained in a part of the ¹¹⁰Cd spectrum and destroyed in certain nonyrast states. In addition, the vibration in the bulk of the low-lying normal states is reportedly preserved. Besides the IBM-1, Harris ^[23] and Mariscotti et al. ^[24] for example, used a framework in which higher-order nuclear angular velocity terms in the cranking model were preserved as well as the moment of inertia was extended to excited bands with two variables (g_o and C). Raduta et al. ^[25] de-quantized a second-order quadrupole boson using a time-dependent variation principle. They applied the coherent state model technique on nuclei, and used the least squares method to calculate the implicated parameters. This paper aims to use the interacting boson model (IBM-1) with a new empirical equation (NEE) to calculate the ground and other states for even-even ^114-128Cd isotopes. The EPS contour was examined for Cd nuclei. In addition, the reduced transition probabilities B(E2) are determined and compared to experimental data.

2. Calculation procedure

For nuclei containing N nucleons, the Interacting Boson Model (IBM) assigns the occupancy to a truncated model space. It provides a quantitative interpretation of indistinguishable particles with angular momentum equal to either 0 or 2 forming pairs. In IBM-1, the Hamiltonian is written as ^[6,26]

The IBM-1 Hamiltonian can be described in nine terms, two of which appear in one-body terms (s and d). The ε_s and ε_d denote the energy of bosons, while the rest are two-body terms (C₀, C ₁, C ₄, v ₀,v ₂ ,u ₀,u ₂). The number of bosons N_b, on the other hand, is conserved. In general, the IBM-1 Hamiltonian in Eq. (1) can also be expressed as ^[28]

where ε = ε_d - ε_s is the boson energy, and the operators are defined as follows:

Total number of d-bosons, pairing, angular momentum, and quadrupole are represented by the operators n^d,P^,L^,Q^ respectively. The T^r operator represents to octupole and hexadecapole as r = 3 and 4, respectively, and H refers to the quadrupole structure parameter. The strength parameters a₀, a₁, a₂, a₃,and a₄ are used to describe the interactions between the bosons. The operator d~ is equal to -1md-m Interaction parameters for the PHINT program are specified: ε = EPS, a₀ = 2 PAIR, a₁ = ELL/2, a₂ = QQ/2, a₃ = 5 OCT, a₄ + 5 HEX, and CHI=0.

The IBM-1 performs three types of dynamic symmetry for nuclei: vibrational U(5), rotational SU(3) and γ-soft O(6), with their eigenvalues given by ^[28].

hence, the energy ε, pairing a₀, and quadrupole a₂ parameters are influence in the U(5), O(6), and SU(3) limits, respectively. Several nuclei have a property that allows them to transition between two or three of the above-mentioned limits. The new empirical equation (NEE) ^[29] was used in this study to calculate the energy levels of Cd isotopes. The energy levels of the GSB are determined depending only on the angular momentum (I) as follows:

This formula has three parameters, A₁, A₂ and A₃. These parameters were obtained after fitting all of the positive experimental (GSB) data. Further to that, the following formula can be used to compute the other bands ^[29]

In addition, the E₀ (I=0) and B can be estimated from the γ-and β-bands.

3. Results and discussion

The results for the energy levels of the ground and other states for ^114-128Cd isotopes, the energy ratios E41+/E21+, the reduced transition probabilities B(E2) results, as well as the BE2;41+⟶21-Cd/BE2;21+⟶01-Cd ratios, and the potential energy surface (EPS), are discussed comprehensively below.

The simplest way to determine the IBM-1 parameters is to use the energy ratio (R) as a starting point for calculations. The energy ratio, R=E41+/E21+, indicates the symmetrical form of a nucleus. The E41+ and E21+ patterns correspond to the 41+ and 21+ energy levels, respectively. It is well understood that R ≈ 3.33 stands for deformed nuclei SU(3), R ≈ 2.5 is for γ-unstable nuclei O(6), and R ≈ 2 is for vibrational nuclei U(5) ^[30-32]. The experimental values of the energy ratio of the Cd isotopes are shown in Table I. The best values for the parameters that provide a suitable fit between the theoretical and experimental energy levels of the Cd isotopes are shown in Table II, while Table III shows the best-fit parameters for the other bands of NEE (6).

The results of the IBM-1 and NEE calculations of the energy levels for ground and other states of the Cd isotope are presented in Figs. 1 and 2, respectively. The low-lying spectrum in Cd nuclei values of the IBM-1 and NEE are really equivalent to available experimental results for 0⁺, 2 ⁺, 4⁺, and 6⁺ states. For the high states, the NEE and IBM-1 calculations are slightly overpredicted with the exception of the ¹¹⁴Cd and 118Cd nuclei, as illustrated in Fig. 1. The levels marked with an asterisk (*) represent cases that the spin or parity of the respective states has not been established experimentally. The IBM-1, NEE, and experimental data on the energy level structures of Cd nuclei appear at first glance to be similar for the other states, as shown in Fig. 2. However, we can render a few predictions based on a comparison of theoretical predictions and experimental results. For the IBM-1 description, the spacing between both 2⁺ states is slightly overestimated in ¹¹⁴Cd and underestimated in ^116-118Cd. There is no gap between both 2⁺ states in ^114-122Cd in the NEE description. Conversely, the comparison between the energies of the states, i.e. for the 6⁺ states, is significantly improved.

Figure 1 Experimental low-lying energies obtained from ^[33] in ^114-128Cd isotopes compared to IBM-1 and NEE calculations. 

Figure 2 The other states of the even-even ^114-128 Cd isotopes. 

As already mentioned, the symmetry shape of a nucleus can be predicted from the energy ratio R=E41+/E21+. Then, the collective dynamics of even-even nuclei energy can be classified into three subgroups: SU(3), O(6), and U(5). The R_4/2 values of low-lying energy levels of Cd isotopes vary as a function of mass number for experimental values, IBM-I, SU(3), O(6), and U(5) limits as presented in Fig. 3. The calculated energy ratios, R_4/2, of Cd nuclei agree well with experimental results, with small exceptions for ^116-120Cd nuclei. Furthermore, this figure clearly demonstrates the transition between the collective structures of U(5) and O(6). We clearly notice that the energy ratio increases very slowly with increasing the neutron number up to ¹²⁰Cd, after which it decreases in the experimental results. However, the IBM-1 results show relatively small fluctuations.

Figure 3 Comparison of the E41+/E21+ values for the experimental data and IBM-1 calculations with the U(5), O(6),and SU(3) limits for Cd isotopes. 

The electromagnetic transitions operator in IBM-1 has a general form ^[7,28]:

where γ₀, α₂, and β (L=0, 1, 2, 3, 4) are parameters identifying the different terms in the corresponding operators. The reduced electrical transition probabilities provide information about nuclei structure. The E2 transition operator is recognized to be a Hermitian tensor. As a result, the N_b is always conserved. Then, the E2 transition operator is represented as ^[34]:

here, the (s^†, d^†) and (s, d) symbols refer to creation and annihilation operators, respectively. The α₂ represents to the effective charge for boson and β₂ is a dimensionless coefficient, β₂ = χ α₂. The B(E2) values for electrical transition probabilities are described in terms of reduced matrix elementsas ^[28,34,35]:

The effective charges e_B for Cd isotopes are shown in Table IV. The experimental values of B(E2:21+⟶01+) were reproduced from the eB values. The results of the IBM-1 and available experimental data for B(E2) values of (g → g, β → β, γ → γ and γ → β) transitions for ^114-128Cd isotopes are shown in Table V. Furthermore, the results of the ground state bands of B(E2) data for 21+⟶01+  41+⟶21+,61+⟶41+,81+⟶61+ levels demonstrated that the IBM-1 calculations with the experimental data for ^114-120Cd isotopes matched with a maximum associated error of 23%, while experimental data are lacking for the other Cd isotopes. The B(E2,21+⟶01+) of^114-118Cd isotopes gradually increases as the number of neutrons increases. For the other isotopes, however, the B(E2) values tend to fluctuate, confirming that these states are expected to be deformed. Moreover, the B(E2,21+⟶21+) decreases with increasing neutron number, except for ¹¹⁴Cd isotope. Other transitional levels exhibit similar variations and can be interpreted similarly. The experimentally published and determined B(E2) values for ^114-128Cd isotopes generally agree well in some places. Other important quantities, such as the B(E2) ratio, are used to demonstrate that the Cd isotopes are deformed nuclei and tend to have a dynamical symmetry U(5)-O(6). For all nuclei under inquiry, the B(E2) ratio is calculated and presented in Fig. 4. The comparison of experimental data and IBM-1 calculations for Cd isotopes is also shown in this figure. We also calculated the BE2;41+⟶21+/BE2;21+⟶01+Cd ratio in Cd isotopes, and found that the B(E2) ratio shows almost little variations with respect to the isotope mass number. These values are smaller than the corresponding γ-unstable nuclei O(6) and approach the vibrational limit U(5). Moreover, the theoretical values of the B(E2) ratio for these nuclei are in good agreement with the experimental data. As a result, Cd isotopes are closer to the U(5)-O(6) boundary.

Figure 4 

The potential energy surface (EPS) application provides information to determine the microscopic and geometric shapes of nuclei. IBM Hamiltonian produced the EPS plots using the Skyrme mean field procedure ^[30]. The IBM-1 energy surface is constructed by combining the IBM-1 Hamiltonian's expectation value with the coherent state N,B,γ^[6]. The creation operators bc† act on a state of boson vacuum |0) to produce the coherent state as follows ^[31,36]:

where

then, the EPS can be written in terms of β and γ as ^[32]:

Where α's parameters are associated with the coefficient of C_L, v₂, v₀, and u₀, as seen in Eq. (1). The term β refers to a nucleus total deformation. Then, the shape of a nucleus could be spherical or distorted depending on whether β = 0 or not. Moreover, the variation in nucleus symmetry is represented by γ term, when γ = 0, the nucleus has a prolate shape; when γ = 60, it has an oblate shape. The deformation energy surfaces of the even-even isotopes ^114-128Cd were estimated as plotted in Fig. 5. The energy surfaces formed a prolate and a slightly oblate threshold that should correlate with proton normal and intruder excitations ^[21]. The ^116-120Cd nuclei, on the other hand, have a deformation shape and a γ-unstable character (γ ≈ 30). While the isotopes ^{114,122,124,126,128}Cd have a potential energy surface that is approximately independent of the triaxial γ-parameter. The U(5)-O(6) transitions were identified in the EPS map for Cd nuclei in the (β, γ) deformation space.

Figure 5 The EPS contour plot for ^114-128Cd nuclei. The color panel represents the EPS values in MeV. 

4. Conclusions

Using the IBM-1 and NEE methods, the ground and other state energies, the electromagnetic transition, and the potential energy surface of ^114-128Cd isotopes were all calculated theoretically. The results of the ground and other energy levels of Cd isotopes are consistent in some places with previous experimental data, i.e., the description of the energies of the 2⁺ state is not well represented in the IBM-1 model. Furthermore, the results of the IBM-1 on reduced transition probabilities B(E2) roughly agree with the available experimental data. Similarly, the BE2;41+⟶21+/BE2;21+⟶01+Cd ratio in Cd isotopes varies relatively minimally with respect to nuclear mass number. The EPS contour results for Cd isotopes demonstrate that the investigated Cd isotopes represent a shape phase change from vibrational U(5) to γ-soft O(6) dynamical symmetry. Finally, the ^114-128Cd nuclei are minimally prolate and exhibit moderate axial distortion.