Vibrational, rotational, and triaxiality features in extended O(6)dynamical symmetry of IBM using three-body interactions

A. M. Khalaf· Azza O. El-Shal· M. M. Taha· M. A. El-Sayed

Abstract The shape transition between the vibrational U(5)and deformed γ-unstable O(6)dynamical symmetries of sd interacting boson model has been investigated by considering a modified O(6) Hamiltonian, providing that the coefficients of the Casimir operator of O(5) are N-dependent, where N is the total number of bosons. The modified O(6) Hamiltonian does not contain the number operator of the d boson, which is responsible for the vibrational motions. In addition, the deformation features can be achieved without using the SU(3) limit by adding to the O(6) dynamical symmetry the three-body interaction［Q QQ］（0）, where Q is the O(6) symmetric quadrupole operator. Moreover, triaxiality can be generated through the inclusion of the cubic d-boson interactionThe classical limit of the potential energy surface (PES),which represents the expected value of the total Hamiltonian in a coherent state, is studied and examined.The modified O(6)model is applied to the eveneven 124-132Xe isotopes. The parameters for the Hamiltonian and the PESs are calculated using a simulated search program to obtain the minimum root mean square deviation between the calculated and experimental excitation energies and B(E2) values for a number of low-lying levels. A

✉M. A. El-Sayed

mathelgohary@yahoo.com

1 Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt

2 Mathematics and Theoretical Physics Department, Nuclear Research Center, Atomic Energy Authority, Cairo 13759,Egypt good agreement between the calculations and experiment results is found.

Keywords Nuclear structure · Extended O(6) of IBM ·Three-body interactions · Coherent state

1 Introduction

The simplest standard version of the interacting boson model (IBM1) [1] has been widely used for describing the collective nuclear quadrupole states observed in medium and heavy nuclei. The building blocks of this model are pairs of correlated nucleons with angular momentum Lπ=0+and 2+, which are represented by s and d bosons,respectively. In its simplest version, the model does not distinguish between proton and neutron bosons.According to this algebraic model, the dynamical symmetries are given by U(5) corresponding to spherical vibrator nuclei,SU(3) corresponding to the axially deformed prolate nuclei, and O(6) corresponding to γ-unstable deformed nuclei.

The shape transitions correspond to break these dynamical symmetries. The critical point symmetry E(5)[2]is designed for the critical point of transition from spherical vibrator U(5) to the deformed γ-unstable O(6).Later, X(5) [3] and Y(5) [4] describe the critical points between the spherical vibrator U(5) and axially deformed prolate rotor SU(3), and between SU(3) and the triaxial deformed shapes, respectively. The correspondence between E(5) in IBM and the solution to the Bohr Hamiltonian in the collective model was studied[5-9],and the existence of an additional prolate-oblate transition was recognized [10]. The U(5)-O(6) shape phase transition based on the concepts of the E(5) critical point dynamical symmetry and the quasi-dynamical symmetry was studied [6-11].

Applying the coherent state formalism [12, 13] to an arbitrary Hamiltonian of the three limiting cases of IBM,as well as to a Hamiltonian represented transition within the region among them,the potential energy surface(PES)can be derived by calculating the expected value of this Hamiltonian, which is related to certain nuclear shapes.Because it is known that IBM1, with only up to two-body interactions, cannot give rise to stable triaxial shapes[14, 15], the inclusion of higher-order interactions in the Hamiltonian, such as three-body interactions between d bosons, must be added [16, 17]. In addition, the effects of three-body interactions in an O(6) Hamiltonian have been considered [18-20]. The collective structure of the nuclei in the A ～130 mass region was discussed within the framework of a rigid triaxial rotor model [21-23] and IBM [1], where the O(6) dynamical symmetry limit was used [19, 24, 25] instead of a geometric γ-unstable rotor model [26]. The even-even xenon nuclei in this mass region A ～130 are soft with regard to the γ-deformation at an almost maximum effective triaxiality with γ ≃30°[27, 28].

The triaxiality in Xe,Ba,and Te nuclei within the mass region A ～130 were studied experimentally [29-33] and interpreted by several nuclear models [34-38]. In this study, we consider the effects of adding terms of cubic dboson interactions and three ［QQQ］interactions(when Q is the O(6) symmetric quadrupole operator) to the symmetry O(6)sdIBM Hamiltonian to generate rotational and triaxial nuclear states.This extended O(6)model with the inclusion of a cubic d-boson interaction, and the inclusion of threebody O(6) symmetric quadrupole terms enables a description of the rotational motion without the use of the SU(3) limit of IBM1 and also allows studying the prolateoblate shape phase transition. The model is applied to the spectroscopy of the even-even xenon isotopes by calculating the PESs using the intrinsic coherent state formalism.

2 O(6) Hamiltonian of interacting boson model with three-body interactions

The Hamiltonian we used is given by the following weighted sum of three terms:

where the first term in the Hamiltonian (1) represents the most general O(6)dynamical symmetry Hamiltonian and is written in a multipole form as follows:

where a0，a1，a3are the model parameters of the Hamiltonian. The operators ^P， ^L, and ^T3are the pairing, angular momentum, and octupole operator, respectively. The explicit expressions for these operators are given as follows:

where θLis a strength parameter. There are five linear independent combinations of type (4), which are determined uniquely by the value of L （L=0，2，3，4，6）. We will choose the single third-order interaction between the d bosons (all θL=0 except θ3), because L=3 is the most effective at creating a triaxial minimum on the potential energy surface. Then, the L=3 cubic d-boson interaction yields the following:

The third term in Hamiltonian (1) is the cubic quadrupole operator and is written in the following form:

with coupling parameter k, and where ^Q is the O(6) symmetric quadrupole operator of the sdIBM given by

3 Boson intrinsic coherent state

The geometrical interpretation of the IBM Hamiltonian can be obtained by introducing an intrinsic coherent state,which allows associating a geometrical shape to it in terms of the deformation parameter β and a departure from axial symmetry γ (β ≥0，0 ≤γ ≤π/3).

The intrinsic coherent state of sdIBM for a nucleus with N valence bosons is given by [12]

where|0〉represents a boson vacuum(inert core)and b†c is a boson creation operator given by the following:

In terms of the parameters β and γ, the expected value of the Hamiltonian is easily obtained from the evaluation of the expected values of each single term.

4 Potential energy surface(PES)and critical points

The PES associated with the classical limit of the Hamiltonian HO(6) is given by its expected value in an intrinsic coherent state (8), yielding the following energy function:

Equation (10) can be rewritten in another form:

where the coefficients A2，A4，c are given by

The PES Eq. (11) is γ-independent and has two independent parameters a0and λ.

The deformed nucleus has the absolute minimum at β/=0. For any stable equilibrium state, the first derivative of E with respect to β must be zero, and the second derivative must be positive.Thus,we obtain the following:

Under the condition a0（N-1）＞λ, the critical point is found at λ=a0（N-1）, and the corresponding E becomes the following:

In Table 1 and Fig. 1, we show the PES calculations corresponding to the modified O(6) limit for different values of N （N =2，4，7，10，12）. In Fig. 1, we show that the critical point of a shape transition depends on the total number of bosons N. We adjusted the PES parameters listed in Table 1 to produce a shape transition at the critical N =7,which for the value N =7 gives a flat β4surface at β=0. Five values of N are presented, one at the critical value of N =7, and two below and two above this value.For N＜7,the nucleus is in a symmetric phase because the PES has a unique minimum at β=0, meaning a spherical shape under equilibrium is obtained. When N increases to the critical point N =7,the non-symmetric and symmetric minima attain the same depth,whereas for N ＞7 the shape at equilibrium is deformed.

In a classical limit （N →∞）, the PES is not explicitly dependent on N and is given by the following:

Table 1 Parameters of the PES for a set of boson numbers N =2，4，7，10，12 (a0 = 0.12 MeV, λ = 0.320 MeV)

Introducing the parameter x such that x=a0/λ, Eq. (17)can be written as follows:

For x＜1,the global minimum is at β=0.For x ＞1,we arrive at a deformed γ-soft shape.For x=1,a second-order phase transition from a spherical to deformed shape occurs(flat β4surface).

Figure 2 shows the evolution of the PESs for various values of x,where a shape phase transition occurs at x=1,providing a flat β4surface close to β=0; that is, the classical limit has the capability of producing a shape transition.

where

The expected value of the cubic d-boson Hamiltonian H3dis obtained using the intrinsic coherent state (8), yielding the following:

where

In addition, the classical potential corresponding to the three-body Hamiltonian HQusing the intrinsic coherent state (8) is given by the following:

where

Adding Eqs.(22)and(24)corresponding to the three-body interactions to Eq. (11), which describes the O(6) dynamical symmetry,yields the total PES of the total Hamiltonian(1).

5 B(E2) ratios

Calculations of the excitation energies and electric quadrupole reduced transition probabilities B(E2)provide a good test for shaping the transition.The electric quadrupole transition operator in the O(6)limit of IBM is given by the following [39]:

with e being the boson effective charge. The reduced electric quadrupole transition probabilities are given by the following:

where Ii and If are the angular momenta for the initial and final states, respectively. For the ground state band, the energy ratios RI/2and the ratios of the E2 transition rates BI+2/2are defined as follows:

The ratios for the U(5)and O(6)limits of IBM are given by the following:

6 Numerical calculations and discussion

To visualize the influence of the cubic boson interaction term on the PES plots, we first represent the PES for the pure O(6)limit Eq.(11)with parameters A2=-2.96，A4=2.64，c=1.4(all in MeV),as shown in Fig.3a.It is known that the shape of the general one- and two-body IBM1 Hamiltonian at equilibrium can never be triaxial. Only the inclusion of specific higher-order boson interaction terms[at least three-body interactions such as H3din Eq. (5)and HQin Eq. (6)] produces triaxiality. The influence of the cubic term H3dof the O(6) limit is studied by plotting the PESs in Fig. 3b-d according to HO(6)+H3dwith the parameters g2=-2.96，g4=-0.32，g6=2.64，a=2.88，c=1.4 (all in MeV). It can be seen that a stable triaxial minimum results at γ=30°and β=0.7.

When the strength parameter of the cubic term H3dequals zero(a=0),a minimum in PES with respect to γ can only occur for γ=0°or γ=60°; that is, the equilibrium shape of the classical limit can never be triaxial.For a/=0,the cubic term lowers the PES with the greatest effect occurring at β0/=0 and γ=30°. Figure 4 illustrates the PESs of EO(6)+E3daccording to Eqs.(11)and(22)in the classical limit.

For the three-body O(6)symmetric quadrupole term HQ,for a fixed λ in Eq. (11) and k/=0 in Eq. (24), the surface energy depends on γ and leads to triaxiality.Figure 5 shows the calculated PESs for k/=0 and various values of the coefficient of the pairing operator a0.

● For a0=0, the minimum potential surface occurs at k ＞0, γ=0, or k＜0, where γ=0°and the PES exhibit spherical (β=0) and deformed (β ＞0) shapes(panel a)

● For a/=0,a spherical shape occurs for a0＜λ(panel b)and a deformed shape occurs for a0＞λ (panels d and e).

● For a0=λ, the spherical shape disappears (panel c)The even-even transitional nuclides124-132Xe represent an excellent example for the extended O(6) triaxial shapes.Because the neutron numbers in this isotopic chain are between N =70 and N =78, the doubly closed shell of Z =50 and N =82 is assumed such that the neutrons are treated as holes, whereas the protons are valance particles when we determine the total number of bosons. For each nucleus, the parameters of the PESs g2，g3，g4，g5，g6，a，g0，γ, which depend on the original parameters of our proposed Hamiltonian,are adjusted from a best fit to the experimental data of the level energies, B(E2), which are the transition probabilities for the ground state bands. A standard χ test is used to conduct the fitting

A good agreement between the present calculations and the experiment results was found. We can see that the collectivity increases smoothly with a decrease in the neutron number from N =78 to N =70, and the value of the R4/2ratio changes from the vibrational limit R4/2≃2 for132Xe to γ-soft rotor R4/2≃2.5 for124Xe.

In Table 4 and Fig. 6, we give the values of the calculated ratios of the E2 transition rates BI+2/2for124，128，132Xe compared to the experimental values and to the calculated O(6) prediction.

The results of the extended O(6)IBM triaxial calculations for the classical limit of the evolution of the PESs for the even-even124-132Xe isotopic chain are shown in Fig.7 as a function of the deformation parameter β2, the PES parameters of which are listed in Table 2.From Table 2,we can see that for132Xe the parameters g3and g5have positive values compared to those of the otherXe isotopes because132Xe shows a quasi-vibrational nucleus demonstrating a minimum PES in the form of a narrow （β，γ）valley from the spherical region （k＜0） to the triaxial region （k ＞0） with γ ≃30°(where k is the strength parameter of the three-body O(6) symmetric quadrupole term). On the other side, the isotope124Xe is a candidate for a deformed O(6) triaxial in a ground state with the shallow minima at γ ≃26°.The resulting contour PESs for124Xe and126Xe are shown in Fig. 8.

7 Conclusion

Table 2 Values of the adopted best PES parameters (in MeV) as derived through the fitting procedure used in the present calculations for a 124-132Xe isotopic chain (where N is the boson number)

Table 3 A comparison between experimental and calculated energy ratios for 124-132Xe isotopic chain

Table 3 A comparison between experimental and calculated energy ratios for 124-132Xe isotopic chain

NB Nuclide R2+2/2+1 R4+1/2+1R3+1/2+1R4+2/2+1R6+1/2+1 124Xe Cal. 2.376 2.471 3.501 4.100 4.316 Exp. 2.391 2.482 3.524 4.061 4.373 7 126Xe Cal. 2.211 2.389 3.398 3.893 4.184 Exp. 2.264 2.423 3.390 3.829 4.207 6 128Xe Cal. 2.141 2.336 3.252 3.699 3.912 Exp. 2.188 2.332 3.227 3.620 3.922 5 130Xe Cal. 2.045 2.311 3.062 3.365 3.651 Exp. 2.093 2.247 3.045 3.373 3.626 4 132Xe Cal. 1.911 2.126 2.753 2.912 3.223 Exp. 1.943 2.157 2.701 2.939 3.162 8

Table 4 Calculated ratios of E2 transition probabilities for 124，128，132Xe isotopes compared to those obtained from the experimental results as well as the prediction of the O(6) limit of IBM

NB Nuclide B4/2 B6/4 B8/6 B10/8 8 124Xe Exp. 1.34(24) 1.59(71) 0.63(29) 0.29(8)Cal. 1.501 1.991 2.490 3.354 O(6) 1.354 1.458 1.420 1.282 6 128Xe Exp. 1.47(2) 1.94(26) 2.39(4) 2.74(114)Cal. 1.790 2.853 4.232 6.001 O(6) 1.309 1.333 1.181 0.897 4 132Xe Exp. 1.24(18)Cal. 2.990 2.032 17.492 O(6) 1.205 1.666 0.625