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

NUCLEAR PHYSICS AND INTERDISCIPLINARY RESEARCH

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

Nuclear Science and Techniques

Vol.31, No.5

Article number 47

Published in print 01 May 2020

Available online 30 Apr 2020

DOI：10.1007/s41365-020-00757-y

43900

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 [QQQ]⁽⁰⁾, where Q is the O(6) symmetric quadrupole operator. Moreover, triaxiality can be generated through the inclusion of the cubic d-boson interaction ${[d^{†} d^{†} d^{†}]}^{(3)} \cdot {[\tilde{d} \tilde{d} \tilde{d}]}^{(3)}$ . The 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 even-even ^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 good agreement between the calculations and experiment results is found.

Nuclear structureExtended O(6) of IBMThree-body interactionsCoherent 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 were 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∼eq30°[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 d-boson 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 three body 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 prolate-oblate 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:

\hat{H} = H_{O(6)} + H_{3 d} + H_{Q},

(1)

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:

H_{O(6)} = a_{0} ({\hat{P}}^{†} \cdot \hat{P}) + a_{1} (\hat{L} \cdot \hat{L}) + a_{3} ({\hat{T}}_{3} \cdot {\hat{T}}_{3}),

(2)

where a₀, a₁, a₃ are the model parameters of the Hamiltonian. The operators $\hat{P}, \hat{L}$ , and ${\hat{T}}_{3}$ are the pairing, angular momentum, and octupole operator, respectively. The explicit expressions for these operators are given as follows:

\begin{array}{l} {\hat{P}}^{†} = & \frac{1}{2} (d^{†} \cdot d^{†} - s^{†} \cdot s^{†}), \\ \hat{L} = & \sqrt{10} {[d^{†} \otimes \tilde{d}]}^{(1)}, \\ {\hat{T}}_{3} = & {[d^{†} \otimes \tilde{d}]}^{(3)} . \end{array}

(3)

Here, s ^$†$ and d ^$†$ are the creation operators of s and d bosons, and $\tilde{d}$ is the annihilation operator of the d boson. The scaler product is defined as ${\hat{T}}_{L} \cdot {\hat{T}}_{L} = \sum_{M} {(- 1)}^{M} T_{L M} T_{L, - M}$ , where TLM corresponds to the M-component of the operator ${\hat{T}}_{L}$ . The operators ${\tilde{d}}_{m} = (- {1)}^{m} d_{- m}$ and $\tilde{s} = s$ are introduced to ensure the correct tensorial characteristic under spatial rotations.

In general, the one- and two-body sdIBM Hamiltonian give rise to spherical, axially symmetric, and γ-unstable shape deformations. There are no stable triaxiality deformed nuclear shapes, unless one includes three-body interactions to break the IBM dynamical symmetries. Thus, to introduce a degree of rotation and triaxiality (γ-dependent), three-body interactions are considered in the second and third terms of Hamiltonian (1).

The second term contains three creation and annihilation operators of the d bosons in a general form as follows:

{\hat{H}}_{3 d} = \sum_{L} θ_{L} {[d^{†} \otimes d^{†} \otimes d^{†}]}^{(L)} \cdot {[\tilde{d} \otimes \tilde{d} \otimes \tilde{d}]}^{(L)}

(4)

where θL is 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:

{\hat{H}}_{3 d} = θ_{3} {[d^{†} \otimes d^{†} \otimes d^{†}]}^{(3)} \cdot {[\tilde{d} \otimes \tilde{d} \otimes \tilde{d}]}^{(3)} .

(5)

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

{\hat{H}}_{Q} = - k {[Q \otimes Q \otimes Q]}^{(0)}

(6)

with coupling parameter k, and where $\hat{Q}$ is the O(6) symmetric quadrupole operator of the sdIBM given by

\hat{Q} = {[d^{†} s + s^{†} \tilde{d}]}^{(2)} .

(7)

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 (β0, 0γπ/3).

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

| C 〉 = \frac{1}{\sqrt{N!}} {(b_{c}^{†})}^{N} | 0 〉,

(8)

where | 0〉 represents a boson vacuum (inert core) and $b_{c}^{†}$ is a boson creation operator given by the following:

b_{c}^{†} = \frac{1}{\sqrt{1 + β^{2}}} [s^{†} + β \cos γ d_{0}^{†} + \frac{1}{\sqrt{2}} \sin γ (d_{2}^{†} + d_{- 2}^{†})] .

(9)

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 H_O(6) is given by its expected value in an intrinsic coherent state (8), yielding the following energy function:

\begin{array}{l} E_{O(6)} (N, β) & = 〈 C | H_{O(6)} | C 〉 \\ = \frac{1}{4} a_{0} N (N - 1) {(\frac{1 - β^{2}}{1 + β^{2}})}^{2} + 6 a_{1} N \frac{β^{2}}{1 + β^{2}} \\ + \frac{7}{5} a_{3} N \frac{β^{2}}{1 + β^{2}} . \end{array}

(10)

Equation (10) can be rewritten in another form:

E_{O(6)} (N, β) = \frac{A_{2} β^{2} + A_{4} β^{4}}{{(1 + β^{2})}^{2}} + c,

(11)

where the coefficients A₂, A₄, c are given by

\begin{array}{l} A_{2} & = λ N - a_{0} N (N - 1), \\ A_{4} & = λ N, \\ c & = \frac{1}{4} a_{0} N (N - 1), \\ with & λ = 6 a_{1} + \frac{7}{5} a_{3} . \end{array}

(12)

The PES Eq. (11) is γ-independent and has two independent parameters a₀ and λ.

To analyze the critical behavior for the energy function Equation (11), the anti-spinodal point occurs when E becomes flat at β = 0 or when ²E/β²|_β=0 = 0 (A₂ = 0), which yields $\frac{a_{0} (N - 1)}{λ} = 1$ .

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:

A_{2} + (2 A_{4} - A_{2}) β^{2} = 0,

(13)

A_{2} + (6 A_{4} - 8 A_{2}) β^{2} - (6 A_{4} - 3 A_{2}) β^{4} > 0.

(14)

Therefore, the equilibrium value of β is as follows:

β_{0} = \pm \sqrt{\frac{A_{2}}{A_{2} - 2 A_{4}}} = \pm \sqrt{\frac{a_{0} (N - 1) - λ}{a_{0} (N - 1) + λ}} .

(15)

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

E_{critical} = \frac{λ N β^{4}}{{(1 + β^{2})}^{2}} + \frac{1}{4} a_{0} N (N - 1) .

(16)

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 β⁴ surface 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.

Fig. 1.

PESs versus the deformation parameter β for the data listed in 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)

N	2	4	7	10	12
A₂	0.4	-0.16	-2.8	-7.6	-12
A₄	0.64	1.28	2.24	3.2	3.84
c	0.06	0.36	1.26	2.70	3.96

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

E (β) = \frac{(λ - a_{0}) β^{2} + λ β^{4}}{{(1 + β^{2})}^{2}} + \frac{1}{4} a_{0} .

(17)

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

\frac{E (β)}{λ} = \frac{(1 - x) β^{2} + β^{4}}{{(1 + β^{2})}^{2}} + \frac{1}{4} x

(18)

: 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 β⁴ surface).

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 β⁴ surface close to β = 0, that is, the classical limit has the capability of producing a shape transition.

Fig. 2.

PESs of the O(6) dynamical symmetry of IBM at the classical limit as a function of the deformation parameter β for (a, b) a spherical shape (x = 0, x = 0.5), (c) a flat β⁴ surface (x = 1), and (d, e) a deformed shape (x = 2, x = 3)

An N dependence occurs if we modify the O(6) Hamiltonian of the sd IBM Equation (2) by providing the coefficients a₁, a₃ of the Casimir operator of O(5) (C₂[O(5)] a₁ L L + a₃T₃ T₃), which are N dependent, that is, a₁ = f₁ + g₁N, a₃ = f₃ + g₃N, and the modified O(6) Hamiltonian in this case becomes the following:

H_{O(6)}^{modified} = a_{0} P^{†} P + (f_{1} + h_{1} N) L \cdot L + (f_{3} + h_{3} N) T_{3} \cdot T_{3} .

(19)

This Hamiltonian is not U(5) invariant but exhibits properties of a spherical vibrator to a high degree of accuracy despite not containing a ${\hat{n}}_{d}$ operator. The PES for this modified O(6) Hamiltonian is also given by Eq. (11) with $λ = (6 f_{1} + \frac{7}{5} f_{3}) + N (6 h_{1} + \frac{7}{5} h_{3})$ .

Equation (11) can be written in the following form:

E (N, β) = \frac{g_{2} β^{2} + g_{4} β^{4} + g_{6} β^{6}}{{(1 + β^{2})}^{3}} + c,

(20)

where

\begin{array}{l} g_{2} = A_{2} = λ N - a_{0} N (N - 1), \\ g_{4} = A_{2} + A_{4} = 2 λ N - a_{0} N (N - 1), \\ g_{6} = A_{4} = λ N . \end{array}

(21)

The expected value of the cubic d-boson Hamiltonian H_3d is obtained using the intrinsic coherent state (8), yielding the following:

\begin{array}{l} E_{3 d} (N, β, γ) & = \frac{1}{7} θ_{3} N (N - 1) (N - 2) \frac{β^{6}}{{(1 + β^{2})}^{3}} \\ \times (- 1 + \cos^{2} 3 γ) \\ = \frac{a β^{6} \cos^{2} 3 γ - a β^{6}}{{(1 + β^{2})}^{3}}, \end{array}

(22)

where

a = \frac{1}{7} θ_{3} N (N - 1) (N - 2) .

(23)

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

\begin{array}{l} E_{Q} (N, β, γ) & = - K \sqrt{\frac{8}{35}} [\frac{3 N (N - 1)}{{(1 + β^{2})}^{3}} + \frac{4 N (N - 1) (N - 2)}{{(1 + β^{2})}^{3}}] \\ \times β^{3} \cos 3 γ \\ = \frac{g_{3} β^{3} \cos 3 γ + g_{5} β^{5} \cos 3 γ}{{(1 + β^{2})}^{3}}, \end{array}

(24)

where

\begin{array}{l} g_{3} & = - K \sqrt{\frac{8}{35}} N (N - 1) (4 N - 5), \\ g_{5} & = - 3 K \sqrt{\frac{8}{35}} N (N - 1) . \end{array}

(25)

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).

\begin{array}{l} E (N, β, γ) = \\ \frac{1}{{(1 + β^{2})}^{3}} [g_{2} β^{2} + g_{3} β^{3} \cos 3 γ + g_{4} β^{4} + g_{5} β^{5} \cos 3 γ \\ + (g_{6} - a) β^{6} + a β^{6} \cos^{2} 3 γ + g_{0}] \end{array}

(26)

This final formula for the PES contains seven parameters, g₂, g₃, g₄, g₅, g₆, a, g₀, in addition to the γ angle.

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]:

\hat{T} (E 2) = e {[d^{†} \times \tilde{s} + s^{†} \times \tilde{d}]}^{(2)}

(27)

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

B (E 2; I_{i} \to I_{f}) = \frac{1}{2 I_{i} + 1} | 〈 I_{f} | | \hat{T} (E 2) | | I_{i} 〉 |^{2}

(28)

where I_i and I_f are the angular momenta for the initial and final states, respectively. For the ground state band, the energy ratios R_I/2 and the ratios of the E2 transition rates B_I+2/2 are defined as follows:

R_{I / 2} = \frac{E (I_{i}^{+})}{E {(2}_{1}^{+})}, B_{I + 2 / 2} = \frac{B (E 2; I + 2 \to I)}{B (E {2; 2}_{1}^{+} \to 0_{1}^{+})} .

(29)

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

R_{I / 2} = {\begin{array}{l} \frac{I}{2} for U(5) \\ \frac{I}{8} (\frac{I}{2} + 3) for O(6) \end{array}},

(30)

B_{I + 2 / 2} = {\begin{array}{l} \frac{1}{2} (I + 2) (1 - \frac{I}{2 N}) for U(5) \\ \frac{5}{2} \frac{(I + 2)}{(I + 5)} (1 - \frac{I}{2 N}) (1 + \frac{I}{2 (N + 4)}) for O(6) \end{array}} .

(31)

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 Equation (11) with parameters A₂ = -2.96, A₄ = 2.64, c = 1.4 (all in MeV), as shown in Figure 3(a). 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 H_3d in Equation (5) and HQ in Equation (6)) produce triaxiality. The influence of the cubic term H_3d of the O(6) limit is studied by plotting the PESs in Figure 3(b–d) according to H_O(6) + H_3d with 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.

Fig. 3.

PES as a function of β

When the strength parameter of the cubic term H_3d equals 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 E_O(6)+E_3d according to Equations (11) and (22) in the classical limit.

Fig. 4.

PESs according to H_O(6)+H_3d in the classical limit as a function of deformation parameters β, demonstrating the influence of the departure from the axial symmetry γ and the strength parameter a of the cubic terms with three creation and three annihilation operators of the d bosons. The parameter a₀ is indicated in the figure, and λ = 4

For the three body O(6) symmetric quadrupole term HQ, for a fixed λ in Equation (11) and k≠0 in Equation (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.

Fig. 5.

PES Equations (11) and (24) in the classical limit as a function of deformation parameter β showing the influence of the strength parameter k of the cubic term interaction [QQQ]⁽⁰⁾ to introduce a degree of triaxiality. The parameter a₀ is indicated in the figure, and λ = 4.

• 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 nuclides ^124-132Xe 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

χ = \sqrt{\frac{1}{n} \sum_{i = 1}^{N} {[\frac{X_{i} (data) - X_{i} (IBM)}{δ X_{i} (data)}]}^{2}},

(32)

where n is the number of experimental data points; Xi(data) and Xi(IBM) are the experimental and calculated spectroscopic properties, respectively; and δXi(data) indicates experimental errors. The experimental data include six energies of levels 2₁, 2₂, 4₁, 3₁, 4₂, 8₁ and five B(E2) reduced quadrupole transition probabilities of the transition $2_{1}^{+} \to 0_{1}^{+} {,4}_{1}^{+} \to 2_{1}^{+} {,2}_{2}^{+} \to 2_{1}^{+} {,2}_{2}^{+} \to 0_{1}^{+} {,2}_{1}^{+} \to 0_{2}^{+}$ and low-spin ground states. The best adopted parameters are listed in Table 2. A comparison between the experimental and our extended O(6)_IBM calculations for the energy ratios $R_{I_{i}^{+} {/ 2}_{1}^{+}} = E (I_{i}^{+}) / E {(2}_{1}^{+})$ and B(E2) ratios B_I+2/2 are shown in Tables 3 and 4. For the pure O(6), the ratios are $R_{2_{2}^{+} {/ 2}_{1}^{+}} = 2.235$ , $R_{4_{1}^{+} {/ 2}_{1}^{+}} = 2.647$ , $R_{3_{1}^{+} {/ 2}_{1}^{+}} = 4.058$ , $R_{4_{2}^{+} {/ 2}_{1}^{+}} = 4.294$ , and $R_{6_{1}^{+} {/ 2}_{1}^{+}} = 4.941$ .

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)

	¹²⁴Xe	¹²⁶Xe	¹²⁸Xe	¹³⁰Xe	¹³²Xe
	N=8	N=7	N=6	N=5	N=4
g₂	1.24	1.26	1.23	1.15	1.05
g₄	3.88	3.57	3.21	2.8	2.001
g₆	4.04	3.36	2.73	2.15	2.101
g₃	-0.158	-0.118	-0.084	-0.056	0.169
g₅	-0.632	-0.472	-0.336	-0.224	0.076
a	2.88	1.8	1.028	0.514	0.205
g₀	1.4	1.05	0.75	0.5	0.22

A comparison between experimental and calculated energy ratios

R_{I_{i}^{+} {/ 2}_{i}^{+}} = E (I_{i}^{+}) / E {(2}_{1}^{+})

for ^124-132Xe isotopic chain

NB	Nuclide	$R_{2_{2}^{+} {/ 2}_{1}^{+}}$	$R_{4_{1}^{+} {/ 2}_{1}^{+}}$	$R_{3_{1}^{+} {/ 2}_{1}^{+}}$	$R_{4_{2}^{+} {/ 2}_{1}^{+}}$	$R_{6_{1}^{+} {/ 2}_{1}^{+}}$
8	¹²⁴Xe
	Cal.	2.376	2.471	3.501	4.100	4.316
	Exp.	2.391	2.482	3.524	4.061	4.373
	Cal.	2.211	2.389	3.398	3.893	4.184
7	¹²⁶Xe
	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

Calculated ratios of E2 transition probabilities

B_{I + 2 / 2} = \frac{B (E 2; I + 2 \to I)}{B (E {2; 2}_{1}^{+} \to 0_{1}^{+})}

I^{π} {= 2}_{1}^{+}

8_{1}^{+}

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	B_4/2	B_6/4	B_8/6	B_10/8
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

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 R_4/2 ratio changes from the vibrational limit R_4/2∼2 for ¹³²Xe to γ-soft rotor R_4/2∼2.5 for ¹²⁴Xe.

In Table 4 and Figure 6, we give the values of the calculated ratios of the E2 transition rates B_I+2/2 for ^124,128,132Xe compared to the experimental values and to the calculated O(6) prediction.

Fig. 6.

Comparison of the calculated

B_{I + 2 / 2} = \frac{B (E 2; I + 2 \to I)}{B (E {2; 2}_{1}^{+} \to 0_{1}^{+})}

ratios of the ground bands in ¹²⁴Xe (NB=8), ¹²⁸Xe (NB=6), and ¹³²Xe (NB=4) compared to the experimental results and O(6) IBM prediction.

The results of the extended O(6)_IBM triaxial calculations for the classical limit of the evolution of the PESs for the even-even ^124-132Xe isotopic chain are shown in Figure 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 for ¹³²Xe the parameters g₃ and g₅ have positive values compared to those of the otherXe isotopes because ¹³²Xe 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 isotope, ¹²⁴Xe is a candidate for a deformed O(6) triaxial in a ground state with the shallow minima at γ∼26°. The resulting contour PESs for ¹²⁴Xe and ¹²⁶Xe are shown in Fig. 8.

Fig. 7.

PES as a function of β for a ^124-132Xe chain.

Fig. 8.

Contour plots of PESs for the nuclei of ¹²⁴Xe and ¹²⁶Xe.

7 Conclusion

The effects of three-body boson interactions were considered. Three d-boson interactions ${[d^{†} d^{†} d^{†}]}^{(3)} \cdot {[\tilde{d} \tilde{d} \tilde{d}]}^{(3)}$ and the cubic [QQQ]⁽⁰⁾ terms, where Q is the O(6) symmetric quadrupole operator, are added to the Hamiltonian of the extended O(6) dynamical symmetry of the IBM. We provided the coefficients of the Casimir operator of O(5), which are N-dependent, where N is the total number of bosons. The modified O(6) model exhibits rotational and triaxiality behaviors. The model is applied to the even-even ^124-132Xe isotopes. A simulated fitting procedure is conducted to obtain the model parameters for each nucleus of the Xe isotopic chain and thus the minimum root mean square deviation between the calculated and experimentally selected set of energy levels and B(E2) transition rates of the yrast states.

References

[1]

F. Iachello and A. Arima, The Interacting Boson Model, Cambridge University Press, Cambridge (1987). https://doi.org/10.1017/CBO9780511895517

[2]

F. Iachelko,

Dynamic Symmetries at the Critical Point

, Phys. Rev. Lett.85(2000)3580. https://doi.org/10.1103/PhysRevLett.85.3580