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

Vibrational, rotational, and triaxiality features in extended O(6) dynamical symmetry of IBM using three-body interactions 1 first-author 2 2 https://orcid.org/0000-0003-4423-9788 2 corresp Corresponding author, mathelgohary@yahoo.com Corresponding author, mathelgohary@yahoo.com mathelgohary@yahoo.com https://orcid.org/0000-0002-6461-4533 Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt Physics Department, Faculty of Science, Al-Azhar University, Cairo, Egypt Mathematics and Theoretical Physics Department, Nuclear Research Center, Atomic Energy Authority, Cairo, P.No. 13759, Egypt Mathematics and Theoretical Physics Department, Nuclear Research Center, Atomic Energy Authority, Cairo, P.No. 13759, Egypt 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 <math id="M1"><mrow><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><msup><mi>d</mi><mo>†</mo></msup><msup><mi>d</mi><mo>†</mo></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mo stretchy="false">[</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup></mrow></math> . 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 124-132 Xe 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 ( E 2) values for a number of low-lying levels. A good agreement between the calculations and experiment results is found. Nuclear structure Extended O(6) of IBM Three-body interactions Coherent state 1 Introduction 1 The simplest standard version of the interacting boson model (IBM1) [ 1 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 L π = 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 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 3 ] and Y(5)[ 4 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 5 - 9 9 5-9 ], and the existence of an additional prolate-oblate transition was recognized [ 10 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 6 - 11 11 6-11 ]. Applying the coherent state formalism [ 12 12 , 13 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 14 , 15 15 ], the inclusion of higher-order interactions in the Hamiltonian, such as three-body interactions between d bosons, must be added [ 16 16 , 17 17 ]. In addition, the effects of three-body interactions in an O(6) Hamiltonian have been considered[ 18 18 - 20 20 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 21 - 23 23 21-23 ] and IBM[ 1 1 ], where the O(6) dynamical symmetry limit was used [ 19 19 , 24 24 , 25 25 ] instead of a geometric γ -unstable rotor model[ 26 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 27 , 28 28 ]. The triaxiality in Xe, Ba, and Te nuclei within the mass region A ∼130 were studied experimentally [ 29 29 - 33 33 29-33 ], and interpreted by several nuclear models [ 34 34 - 38 38 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 1 The Hamiltonian we used is given by the following weighted sum of three terms: <math display="block" id="M2"><mrow><mover accent="true"><mi>H</mi><mo>^</mo></mover><mo>=</mo><msub><mi>H</mi><mrow><mtext>O(6)</mtext></mrow></msub><mo>+</mo><msub><mi>H</mi><mrow><mn>3</mn><mi>d</mi></mrow></msub><mo>+</mo><msub><mi>H</mi><mi>Q</mi></msub><mn>,</mn></mrow></math> 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: <math display="block" id="M3"><mrow><msub><mi>H</mi><mrow><mtext>O(6)</mtext></mrow></msub><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msup><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>†</mo></msup><mo>⋅</mo><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>⋅</mo><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>+</mo><msub><mi>a</mi><mn>3</mn></msub><mo stretchy="false">(</mo><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>⋅</mo><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where a 0 , a 1 , a 3 are the model parameters of the Hamiltonian. The operators <math id="M4"><mrow><mover accent="true"><mi>P</mi><mo>^</mo></mover><mn>,</mn><mover accent="true"><mi>L</mi><mo>^</mo></mover></mrow></math> , and <math id="M5"><mrow><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub></mrow></math> are the pairing, angular momentum, and octupole operator, respectively. The explicit expressions for these operators are given as follows: <math display="block" id="M6"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msup><mover accent="true"><mi>P</mi><mo>^</mo></mover><mo>†</mo></msup><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msup><mi>d</mi><mo>†</mo></msup><mo>⋅</mo><msup><mi>d</mi><mo>†</mo></msup><mo>−</mo><msup><mi>s</mi><mo>†</mo></msup><mo>⋅</mo><msup><mi>s</mi><mo>†</mo></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mover accent="true"><mi>L</mi><mo>^</mo></mover><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msqrt><mrow><mn>10</mn></mrow></msqrt><msup><mrow><mo stretchy="false">[</mo><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mn>3</mn></msub><mo>=</mo></mrow></mtd><mtd columnalign="left"><mrow><msup><mrow><mo stretchy="false">[</mo><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo stretchy="false">]</mo></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> Here, s <math id="M7"><mo>†</mo></math> and d <math id="M8"><mo>†</mo></math> are the creation operators of s and d bosons, and <math id="M9"><mover accent="true"><mi>d</mi><mo>˜</mo></mover></math> is the annihilation operator of the d boson. The scaler product is defined as <math id="M10"><mrow><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>L</mi></msub><mo>⋅</mo><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>L</mi></msub><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>M</mi></munder></mstyle><msup><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>M</mi></msup><msub><mi>T</mi><mrow><mi>L</mi><mi>M</mi></mrow></msub><msub><mi>T</mi><mrow><mi>L</mi><mn>,</mn><mo>−</mo><mi>M</mi></mrow></msub></mrow></math> , where T LM corresponds to the M-component of the operator <math id="M11"><mrow><msub><mover accent="true"><mi>T</mi><mo>^</mo></mover><mi>L</mi></msub></mrow></math> . The operators <math id="M12"><mrow><msub><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mi>m</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><msup><mrow><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>m</mi></msup><msub><mi>d</mi><mrow><mo>−</mo><mi>m</mi></mrow></msub></mrow></math> and <math id="M13"><mrow><mover accent="true"><mi>s</mi><mo>˜</mo></mover><mo>=</mo><mi>s</mi></mrow></math> 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: <math display="block" id="M14"><mrow><msub><mover accent="true"><mi>H</mi><mo>^</mo></mover><mrow><mn>3</mn><mi>d</mi></mrow></msub><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>L</mi></munder></mstyle><msub><mi>θ</mi><mi>L</mi></msub><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><msup><mi>d</mi><mo>†</mo></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mrow><mo>[</mo><mrow><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow></msup></mrow></math> where θ L 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 θ L =0 except θ 3 ), 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: <math display="block" id="M15"><mrow><msub><mover accent="true"><mi>H</mi><mo>^</mo></mover><mrow><mn>3</mn><mi>d</mi></mrow></msub><mo>=</mo><msub><mi>θ</mi><mn>3</mn></msub><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><msup><mi>d</mi><mo>†</mo></msup><mo>⊗</mo><msup><mi>d</mi><mo>†</mo></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mrow><mo>[</mo><mrow><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mo>⊗</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mn>.</mn></mrow></math> The third term in Hamiltonian (1) is the cubic quadrupole operator and is written in the following form: <math display="block" id="M16"><mrow><msub><mover accent="true"><mi>H</mi><mo>^</mo></mover><mi>Q</mi></msub><mo>=</mo><mo>−</mo><mi>k</mi><msup><mrow><mrow><mo>[</mo><mrow><mi>Q</mi><mo>⊗</mo><mi>Q</mi><mo>⊗</mo><mi>Q</mi></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msup></mrow></math> with coupling parameter k , and where <math id="M17"><mover accent="true"><mi>Q</mi><mo>^</mo></mover></math> is the O(6) symmetric quadrupole operator of the sdIBM given by <math display="block" id="M18"><mrow><mover accent="true"><mi>Q</mi><mo>^</mo></mover><mo>=</mo><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><mi>s</mi><mo>+</mo><msup><mi>s</mi><mo>†</mo></msup><mover accent="true"><mi>d</mi><mo>˜</mo></mover></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup><mn>.</mn></mrow></math> 3 Boson Intrinsic Coherent State. 1 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 12 ] <math display="block" id="M19"><mrow><mo>|</mo><mi>C</mi><mtext> </mtext><mo>〉</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mi>N</mi><mo>!</mo></mrow></msqrt></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mi>b</mi><mi>c</mi><mo>†</mo></msubsup></mrow><mo>)</mo></mrow></mrow><mi>N</mi></msup><mo>|</mo><mn>0</mn><mo>〉</mo><mn>,</mn></mrow></math> where | 0〉 represents a boson vacuum (inert core) and <math id="M20"><mrow><msubsup><mi>b</mi><mtext>c</mtext><mo>†</mo></msubsup></mrow></math> is a boson creation operator given by the following: <math display="block" id="M21"><mrow><msubsup><mi>b</mi><mtext>c</mtext><mo>†</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></msqrt></mrow></mfrac><mrow><mo>[</mo><mrow><msup><mi>s</mi><mo>†</mo></msup><mo>+</mo><mi>β</mi><mi>cos</mi><mi>γ</mi><msubsup><mi>d</mi><mn>0</mn><mo>†</mo></msubsup><mo>+</mo><mfrac><mn>1</mn><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac><mi>sin</mi><mi>γ</mi><mo stretchy="false">(</mo><msubsup><mi>d</mi><mn>2</mn><mo>†</mo></msubsup><mo>+</mo><msubsup><mi>d</mi><mrow><mo>−</mo><mn>2</mn></mrow><mo>†</mo></msubsup><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> 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 1 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: <math display="block" id="M22"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>E</mi><mrow><mtext>O(6)</mtext></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mrow><mo>〈</mo><mrow><mi>C</mi><mo>|</mo><msub><mi>H</mi><mrow><mtext>O(6)</mtext></mrow></msub><mo>|</mo><mi>C</mi></mrow><mo>〉</mo></mrow></mrow></mtd><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>β</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>+</mo><mn>6</mn><msub><mi>a</mi><mn>1</mn></msub><mi>N</mi><mfrac><mrow><msup><mi>β</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></mfrac></mrow></mtd><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mfrac><mn>7</mn><mn>5</mn></mfrac><msub><mi>a</mi><mn>3</mn></msub><mi>N</mi><mfrac><mrow><msup><mi>β</mi><mn>2</mn></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup></mrow></mfrac><mn>.</mn></mrow></mtd><mtd columnalign="left"><mrow/></mtd></mtr></mtable></mrow></math> Equation (10) can be rewritten in another form: <math display="block" id="M23"><mrow><msub><mi>E</mi><mrow><mtext>O(6)</mtext></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>A</mi><mn>2</mn></msub><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msub><mi>A</mi><mn>4</mn></msub><msup><mi>β</mi><mn>4</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mn>,</mn></mrow></math> where the coefficients A 2 , A 4 , c are given by <math display="block" id="M24"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>λ</mi><mi>N</mi><mo>−</mo><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>A</mi><mn>4</mn></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mi>λ</mi><mi>N</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mi>c</mi></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext>with</mtext></mrow></mtd><mtd columnalign="left"><mrow><mi>λ</mi><mo>=</mo><mn>6</mn><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>7</mn><mn>5</mn></mfrac><msub><mi>a</mi><mn>3</mn></msub><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The PES Eq. (11) is γ -independent and has two independent parameters a 0 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 2 E / β 2 | β =0 = 0 ( A 2 = 0), which yields <math id="M25"><mrow><mfrac><mrow><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mi>λ</mi></mfrac><mo>=</mo><mn>1</mn></mrow></math> . 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: <math display="block" id="M26"><mrow><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>2</mn><msub><mi>A</mi><mn>4</mn></msub><mo>−</mo><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><msup><mi>β</mi><mn>2</mn></msup><mo>=</mo><mn>0,</mn></mrow></math> <math display="block" id="M27"><mrow><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mn>6</mn><msub><mi>A</mi><mn>4</mn></msub><mo>−</mo><mn>8</mn><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><msup><mi>β</mi><mn>2</mn></msup><mo>−</mo><mo stretchy="false">(</mo><mn>6</mn><msub><mi>A</mi><mn>4</mn></msub><mo>−</mo><mn>3</mn><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo><msup><mi>β</mi><mn>4</mn></msup><mo>></mo><mn>0.</mn></mrow></math> Therefore, the equilibrium value of β is as follows: <math display="block" id="M28"><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>=</mo><mo>±</mo><msqrt><mrow><mfrac><mrow><msub><mi>A</mi><mn>2</mn></msub></mrow><mrow><msub><mi>A</mi><mn>2</mn></msub><mo>−</mo><mn>2</mn><msub><mi>A</mi><mn>4</mn></msub></mrow></mfrac></mrow></msqrt><mo>=</mo><mo>±</mo><msqrt><mrow><mfrac><mrow><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>λ</mi></mrow><mrow><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>λ</mi></mrow></mfrac></mrow></msqrt><mn>.</mn></mrow></math> Under the condition a 0 (N-1) > λ , the critical point is found at λ = a 0 ( N -1), and the corresponding E becomes the following: <math display="block" id="M29"><mrow><msub><mi>E</mi><mrow><mtext>critical</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>N</mi><msup><mi>β</mi><mn>4</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In Table 1 and Fig. 1 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 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 Table 1 to produce a shape transition at the critical N = 7, which for the value N = 7 gives a flat β 4 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. 1 1 N 1 1 2 1 1 4 1 1 7 1 1 10 1 1 12 1 1 A 2 1 1 0.4 1 1 -0.16 1 1 -2.8 1 1 -7.6 1 1 -12 1 1 A 4 1 1 0.64 1 1 1.28 1 1 2.24 1 1 3.2 1 1 3.84 1 1 c 1 1 0.06 1 1 0.36 1 1 1.26 1 1 2.70 1 1 3.96 In a classical limit ( N →∞), the PES is not explicitly dependent on N and is given by the following: <math display="block" id="M30"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mi>λ</mi><mo>−</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">)</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><mi>λ</mi><msup><mi>β</mi><mn>4</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mn>.</mn></mrow></math> Introducing the parameter x such that x = a 0 / λ , Equation (17) can be written as follows: <math display="block" id="M31"><mrow><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><mi>λ</mi></mfrac><mo>=</mo><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>4</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>x</mi></mrow></math> : 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 β 4 surface). Figure 2 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 β 4 surface close to β = 0, that is, the classical limit has the capability of producing a shape transition. An N dependence occurs if we modify the O(6) Hamiltonian of the sd IBM Equation (2) by providing the coefficients a 1 , a 3 of the Casimir operator of O(5) ( C 2 [O(5)] a 1 L L + a 3 T 3 T 3 ), which are N dependent, that is, a 1 = f 1 + g 1 N , a 3 = f 3 + g 3 N , and the modified O(6) Hamiltonian in this case becomes the following: <math display="block" id="M32"><mrow><msubsup><mi>H</mi><mrow><mtext>O(6)</mtext></mrow><mrow><mtext>modified</mtext></mrow></msubsup><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><msup><mi>P</mi><mo>†</mo></msup><mi>P</mi><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><msub><mi>h</mi><mn>1</mn></msub><mi>N</mi></mrow><mo>)</mo></mrow><mi>L</mi><mo>⋅</mo><mi>L</mi><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>f</mi><mn>3</mn></msub><mo>+</mo><msub><mi>h</mi><mn>3</mn></msub><mi>N</mi></mrow><mo>)</mo></mrow><msub><mi>T</mi><mn>3</mn></msub><mo>⋅</mo><msub><mi>T</mi><mn>3</mn></msub><mn>.</mn></mrow></math> This Hamiltonian is not U(5) invariant but exhibits properties of a spherical vibrator to a high degree of accuracy despite not containing a <math id="M33"><mrow><msub><mover accent="true"><mi>n</mi><mo>^</mo></mover><mi>d</mi></msub></mrow></math> operator. The PES for this modified O(6) Hamiltonian is also given by Eq. (11) with <math id="M34"><mrow><mi>λ</mi><mo>=</mo><mrow><mo>(</mo><mrow><mn>6</mn><msub><mi>f</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>7</mn><mn>5</mn></mfrac><msub><mi>f</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow><mo>+</mo><mi>N</mi><mrow><mo>(</mo><mrow><mn>6</mn><msub><mi>h</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>7</mn><mn>5</mn></mfrac><msub><mi>h</mi><mn>3</mn></msub></mrow><mo>)</mo></mrow></mrow></math> . Equation (11) can be written in the following form: <math display="block" id="M35"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>g</mi><mn>2</mn></msub><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msub><mi>g</mi><mn>4</mn></msub><msup><mi>β</mi><mn>4</mn></msup><mo>+</mo><msub><mi>g</mi><mn>6</mn></msub><msup><mi>β</mi><mn>6</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mi>c</mi><mn>,</mn></mrow></math> where <math display="block" id="M36"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>g</mi><mn>2</mn></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>=</mo><mi>λ</mi><mi>N</mi><mo>−</mo><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>g</mi><mn>4</mn></msub><mo>=</mo><msub><mi>A</mi><mn>2</mn></msub><mo>+</mo><msub><mi>A</mi><mn>4</mn></msub><mo>=</mo><mn>2</mn><mi>λ</mi><mi>N</mi><mo>−</mo><msub><mi>a</mi><mn>0</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><msub><mi>g</mi><mn>6</mn></msub><mo>=</mo><msub><mi>A</mi><mn>4</mn></msub><mo>=</mo><mi>λ</mi><mi>N</mi><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> The expected value of the cubic d-boson Hamiltonian H 3 d is obtained using the intrinsic coherent state (8), yielding the following: <math display="block" id="M37"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>E</mi><mrow><mn>3</mn><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mn>,</mn><mi>γ</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><msub><mi>θ</mi><mn>3</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>β</mi><mn>6</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>+</mo><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><mn>3</mn><mi>γ</mi><mo stretchy="false">)</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><mi>a</mi><msup><mi>β</mi><mn>6</mn></msup><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><mn>3</mn><mi>γ</mi><mo>−</mo><mi>a</mi><msup><mi>β</mi><mn>6</mn></msup></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where <math display="block" id="M38"><mrow><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>7</mn></mfrac><msub><mi>θ</mi><mn>3</mn></msub><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></math> In addition, the classical potential corresponding to the three-body Hamiltonian H Q using the intrinsic coherent state (8) is given by the following: <math display="block" id="M39"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>E</mi><mi>Q</mi></msub><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mn>,</mn><mi>γ</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo>−</mo><mi>K</mi><msqrt><mrow><mfrac><mn>8</mn><mrow><mn>35</mn></mrow></mfrac></mrow></msqrt><mrow><mo>[</mo><mrow><mfrac><mrow><mn>3</mn><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>4</mn><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>×</mo><msup><mi>β</mi><mn>3</mn></msup><mi>cos</mi><mn>3</mn><mi>γ</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>=</mo><mfrac><mrow><msub><mi>g</mi><mn>3</mn></msub><msup><mi>β</mi><mn>3</mn></msup><mi>cos</mi><mn>3</mn><mi>γ</mi><mo>+</mo><msub><mi>g</mi><mn>5</mn></msub><msup><mi>β</mi><mn>5</mn></msup><mi>cos</mi><mn>3</mn><mi>γ</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where <math display="block" id="M40"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>g</mi><mn>3</mn></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo>−</mo><mi>K</mi><msqrt><mrow><mfrac><mn>8</mn><mrow><mn>35</mn></mrow></mfrac></mrow></msqrt><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>4</mn><mi>N</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>g</mi><mn>5</mn></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><mo>−</mo><mn>3</mn><mi>K</mi><msqrt><mrow><mfrac><mn>8</mn><mrow><mn>35</mn></mrow></mfrac></mrow></msqrt><mi>N</mi><mo stretchy="false">(</mo><mi>N</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mn>.</mn></mrow></mtd></mtr></mtable></mrow></math> 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). <math display="block" id="M41"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>N</mi><mn>,</mn><mi>β</mi><mn>,</mn><mi>γ</mi><mo stretchy="false">)</mo><mo>=</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>g</mi><mn>2</mn></msub><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msub><mi>g</mi><mn>3</mn></msub><msup><mi>β</mi><mn>3</mn></msup><mi>cos</mi><mn>3</mn><mi>γ</mi><mo>+</mo><msub><mi>g</mi><mn>4</mn></msub><msup><mi>β</mi><mn>4</mn></msup><mo>+</mo><msub><mi>g</mi><mn>5</mn></msub><msup><mi>β</mi><mn>5</mn></msup><mi>cos</mi><mn>3</mn><mi>γ</mi></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mo>+</mo><mrow><mo>(</mo><mrow><msub><mi>g</mi><mn>6</mn></msub><mo>−</mo><mi>a</mi></mrow><mo>)</mo></mrow><msup><mi>β</mi><mn>6</mn></msup><mo>+</mo><mi>a</mi><msup><mi>β</mi><mn>6</mn></msup><msup><mstyle mathsize="140%" displaystyle="true"><mrow><mi>cos</mi></mrow></mstyle><mn>2</mn></msup><mn>3</mn><mi>γ</mi><mo>+</mo><msub><mi>g</mi><mn>0</mn></msub><mo stretchy="false">]</mo></mrow></mtd></mtr></mtable></mrow></math> This final formula for the PES contains seven parameters, g 2 , g 3 , g 4 , g 5 , g 6 , a , g 0 , in addition to the γ angle. 5 B(E2) Ratios 1 Calculations of the excitation energies and electric quadrupole reduced transition probabilities B ( E 2) 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 39 ]: <math display="block" id="M42"><mrow><mover accent="true"><mi>T</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>e</mi><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><mo>×</mo><mover accent="true"><mi>s</mi><mo>˜</mo></mover><mo>+</mo><msup><mi>s</mi><mo>†</mo></msup><mo>×</mo><mover accent="true"><mi>d</mi><mo>˜</mo></mover></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msup></mrow></math> with e being the boson effective charge. The reduced electric quadrupole transition probabilities are given by the following: <math display="block" id="M43"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo>;</mo><msub><mi>I</mi><mtext>i</mtext></msub><mo>→</mo><msub><mi>I</mi><mtext>f</mtext></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><msub><mi>I</mi><mtext>i</mtext></msub><mo>+</mo><mn>1</mn></mrow></mfrac><mo>|</mo><mo>〈</mo><msub><mi>I</mi><mtext>f</mtext></msub><mo>|</mo><mo>|</mo><mover accent="true"><mi>T</mi><mo>^</mo></mover><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo stretchy="false">)</mo><mo>|</mo><mo>|</mo><msub><mi>I</mi><mtext>i</mtext></msub><mo>〉</mo><msup><mo>|</mo><mn>2</mn></msup></mrow></math> 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 E 2 transition rates B I +2/2 are defined as follows: <math display="block" id="M44"><mrow><msub><mi>R</mi><mrow><mi>I</mi><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>E</mi><mo stretchy="false">(</mo><msubsup><mi>I</mi><mtext>i</mtext><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow><mrow><mi>E</mi><msubsup><mrow><mo stretchy="false">(</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mn>,</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub><mi>B</mi><mrow><mi>I</mi><mo>+</mo><mn>2</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2</mn><mo>;</mo><mi>I</mi><mo>+</mo><mn>2</mn><mo>→</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><msubsup><mrow><mn>2</mn><mo>;</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> The ratios for the U(5) and O(6) limits of IBM are given by the following: <math display="block" id="M45"><mrow><msub><mi>R</mi><mrow><mi>I</mi><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mi>I</mi><mn>2</mn></mfrac><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>for</mtext><mtext> </mtext><mtext>U(5)</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mi>I</mi><mn>8</mn></mfrac><mrow><mo>(</mo><mrow><mfrac><mi>I</mi><mn>2</mn></mfrac><mo>+</mo><mn>3</mn></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>for</mtext><mtext> </mtext><mtext>O(6)</mtext></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M46"><mrow><msub><mi>B</mi><mrow><mi>I</mi><mo>+</mo><mn>2</mn><mo>/</mo><mn>2</mn></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mrow><mi>I</mi><mo>+</mo><mn>2</mn></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mi>I</mi><mrow><mn>2</mn><mi>N</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext>for</mtext><mtext> </mtext><mtext>U(5)</mtext></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd><mtd columnalign="left"><mrow><mfrac><mn>5</mn><mn>2</mn></mfrac><mfrac><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>5</mn><mo stretchy="false">)</mo></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mi>I</mi><mrow><mn>2</mn><mi>N</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mi>I</mi><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mtext> </mtext><mtext> </mtext><mtext>for</mtext><mtext> </mtext><mtext>O(6)</mtext></mrow></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mn>.</mn></mrow></math> 6 Numerical Calculations and Discussion 1 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 A 2 = -2.96, A 4 = 2.64, c = 1.4 (all in MeV), as shown in Figure 3(a) 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 3 d in Equation (5) and H Q in Equation (6)) produce triaxiality. The influence of the cubic term H 3 d of the O(6) limit is studied by plotting the PESs in Figure 3(b–d) Figure 3(b–d) according to H O(6) + H 3 d with the parameters g 2 = -2.96, g 4 = -0.32, g 6 = 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 H 3 d 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 β 0 ≠0 and γ = 30°. Figure (4) Figure (4) illustrates the PESs of E O(6) + E 3 d according to Equations (11) and (22) in the classical limit. For the three body O(6) symmetric quadrupole term H Q , 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) Figure (5) shows the calculated PESs for k ≠0 and various values of the coefficient of the pairing operator a 0 . • For a 0 = 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 a 0 < λ (panel b) and a deformed shape occurs for a 0 > λ (panels d and e). • For a 0 = λ , the spherical shape disappears (panel c) The even-even transitional nuclides 124-132 Xe 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 g 2 , g 3 , g 4 , g 5 , g 6 , a , g 0 , 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 ( E 2), which are the transition probabilities for the ground state bands. A standard χ test is used to conduct the fitting <math display="block" id="M47"><mrow><mi>χ</mi><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mi>n</mi></mfrac><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover></mstyle><msup><mrow><mrow><mo>[</mo><mrow><mfrac><mrow><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mtext>data</mtext><mo stretchy="false">)</mo><mo>−</mo><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mtext>IBM</mtext><mo stretchy="false">)</mo></mrow><mrow><mi>δ</mi><msub><mi>X</mi><mi>i</mi></msub><mo stretchy="false">(</mo><mtext>data</mtext><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow></mrow><mn>2</mn></msup></mrow></msqrt><mn>,</mn></mrow></math> where n is the number of experimental data points; X i (data) and X i (IBM) are the experimental and calculated spectroscopic properties, respectively; and δX i (data) indicates experimental errors. The experimental data include six energies of levels 2 1 , 2 2 , 4 1 , 3 1 , 4 2 , 8 1 and five B ( E 2) reduced quadrupole transition probabilities of the transition <math id="M48"><mrow><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mn>,4</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mn>,2</mn></mrow><mn>2</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mn>,2</mn></mrow><mn>2</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mn>,2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo>→</mo><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></mrow></math> and low-spin ground states. The best adopted parameters are listed in Table 2 Table 2 . A comparison between the experimental and our extended O(6) IBM calculations for the energy ratios <math id="M49"><mrow><msub><mi>R</mi><mrow><msubsup><mi>I</mi><mi>i</mi><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><msubsup><mi>I</mi><mi>i</mi><mo>+</mo></msubsup><mo stretchy="false">)</mo><mo>/</mo><mi>E</mi><msubsup><mrow><mo stretchy="false">(</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup><mo stretchy="false">)</mo></mrow></math> and B ( E 2) ratios B I +2/2 are shown in Tables 3 Tables 3 and 4 4 . For the pure O(6), the ratios are <math id="M50"><mrow><msub><mi>R</mi><mrow><msubsup><mn>2</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mn>2.235</mn></mrow></math> , <math id="M51"><mrow><msub><mi>R</mi><mrow><msubsup><mn>4</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mn>2.647</mn></mrow></math> , <math id="M52"><mrow><msub><mi>R</mi><mrow><msubsup><mn>3</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mn>4.058</mn></mrow></math> , <math id="M53"><mrow><msub><mi>R</mi><mrow><msubsup><mn>4</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mn>4.294</mn></mrow></math> , and <math id="M54"><mrow><msub><mi>R</mi><mrow><msubsup><mn>6</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub><mo>=</mo><mn>4.941</mn></mrow></math> . 1 1 1 1 124 Xe 1 1 126 Xe 1 1 128 Xe 1 1 130 Xe 1 1 132 Xe 1 1   1 1 N =8 1 1 N =7 1 1 N =6 1 1 N =5 1 1 N =4 1 1 g 2 1 1 1.24 1 1 1.26 1 1 1.23 1 1 1.15 1 1 1.05 1 1 g 4 1 1 3.88 1 1 3.57 1 1 3.21 1 1 2.8 1 1 2.001 1 1 g 6 1 1 4.04 1 1 3.36 1 1 2.73 1 1 2.15 1 1 2.101 1 1 g 3 1 1 -0.158 1 1 -0.118 1 1 -0.084 1 1 -0.056 1 1 0.169 1 1 g 5 1 1 -0.632 1 1 -0.472 1 1 -0.336 1 1 -0.224 1 1 0.076 1 1 a 1 1 2.88 1 1 1.8 1 1 1.028 1 1 0.514 1 1 0.205 1 1 g 0 1 1 1.4 1 1 1.05 1 1 0.75 1 1 0.5 1 1 0.22 1 1 NB 1 1 Nuclide 1 1 <math id="M56"><mrow><msub><mi>R</mi><mrow><msubsup><mn>2</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub></mrow></math> 1 1 <math id="M57"><mrow><msub><mi>R</mi><mrow><msubsup><mn>4</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub></mrow></math> 1 1 <math id="M58"><mrow><msub><mi>R</mi><mrow><msubsup><mn>3</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub></mrow></math> 1 1 <math id="M59"><mrow><msub><mi>R</mi><mrow><msubsup><mn>4</mn><mn>2</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub></mrow></math> 1 1 <math id="M60"><mrow><msub><mi>R</mi><mrow><msubsup><mn>6</mn><mn>1</mn><mo>+</mo></msubsup><msubsup><mrow><mo>/</mo><mn>2</mn></mrow><mn>1</mn><mo>+</mo></msubsup></mrow></msub></mrow></math> 1 1 8 1 1 124 Xe 1 1 1 1 Cal. 1 1 2.376 1 1 2.471 1 1 3.501 1 1 4.100 1 1 4.316 1 1   1 1 Exp. 1 1 2.391 1 1 2.482 1 1 3.524 1 1 4.061 1 1 4.373 1 1 1 1 Cal. 1 1 2.211 1 1 2.389 1 1 3.398 1 1 3.893 1 1 4.184 1 1 7 1 1 126 Xe 1 1   1 1 Exp. 1 1 2.264 1 1 2.423 1 1 3.390 1 1 3.829 1 1 4.207 1 1 6 1 1 128 Xe 1 1 1 1 Cal. 1 1 2.141 1 1 2.336 1 1 3.252 1 1 3.699 1 1 3.912 1 1   1 1 Exp. 1 1 2.188 1 1 2.332 1 1 3.227 1 1 3.620 1 1 3.922 1 1 5 1 1 130 Xe 1 1 1 1 Cal. 1 1 2.045 1 1 2.311 1 1 3.062 1 1 3.365 1 1 3.651 1 1   1 1 Exp. 1 1 2.093 1 1 2.247 1 1 3.045 1 1 3.373 1 1 3.626 1 1 4 1 1 132 Xe 1 1 1 1 Cal. 1 1 1.911 1 1 2.126 1 1 2.753 1 1 2.912 1 1 3.223 1 1   1 1 Exp. 1 1 1.943 1 1 2.157 1 1 2.701 1 1 2.939 1 1 3.162 1 1 NB 1 1 Nuclide 1 1 B 4/2 1 1 B 6/4 1 1 B 8/6 1 1 B 10/8 1 1 8 1 1 124 Xe 1 1 1 1 Exp. 1 1 1.34(24) 1 1 1.59(71) 1 1 0.63(29) 1 1 0.29(8) 1 1 1 1 Cal. 1 1 1.501 1 1 1.991 1 1 2.490 1 1 3.354 1 1 1 1 O(6) 1 1 1.354 1 1 1.458 1 1 1.420 1 1 1.282 1 1 6 1 1 128 Xe 1 1 1 1 Exp. 1 1 1.47(2) 1 1 1.94(26) 1 1 2.39(4) 1 1 2.74(114) 1 1 1 1 Cal. 1 1 1.790 1 1 2.853 1 1 4.232 1 1 6.001 1 1 1 1 O(6) 1 1 1.309 1 1 1.333 1 1 1.181 1 1 0.897 1 1 4 1 1 132 Xe 1 1 1 1 Exp. 1 1 1.24(18) 1 1 1 1 1 1 1 1 1 1 Cal. 1 1 2.990 1 1 2.032 1 1 17.492 1 1 1 1 1 1 O(6) 1 1 1.205 1 1 1.666 1 1 0.625 1 1 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 132 Xe to γ -soft rotor R 4/2 ∼2.5 for 124 Xe. In Table 4 and Figure 6 Figure 6 , we give the values of the calculated ratios of the E 2 transition rates B I +2/2 for 124,128,132 Xe 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-even 124-132 Xe isotopic chain are shown in Figure 7 Figure 7 as a function of the deformation parameter β 2 , the PES parameters of which are listed in Table 2 Table 2 . From Table 2, we can see that for 132 Xe the parameters g 3 and g 5 have positive values compared to those of the otherXe isotopes because 132 Xe 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, 124 Xe 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 124 Xe and 126 Xe are shown in Fig. 8 Fig. 8 . 7 Conclusion 1 The effects of three-body boson interactions were considered. Three d -boson interactions <math id="M65"><mrow><msup><mrow><mrow><mo>[</mo><mrow><msup><mi>d</mi><mo>†</mo></msup><msup><mi>d</mi><mo>†</mo></msup><msup><mi>d</mi><mo>†</mo></msup></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup><mo>⋅</mo><msup><mrow><mrow><mo>[</mo><mrow><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mover accent="true"><mi>d</mi><mo>˜</mo></mover><mover accent="true"><mi>d</mi><mo>˜</mo></mover></mrow><mo>]</mo></mrow></mrow><mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msup></mrow></math> 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. 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(Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-020-00757-y http://dx.doi.org/10.1007/s41365-020-00757-y . 2020 Nuclear Science and Techniques 5 31