Nuclear alternating-parity bands and transition rates in a model of coherent quadrupole-octupole motion in neutron-rich barium isotopes

Nuclear alternating-parity bands and transition rates in a model of coherent quadrupole-octupole motion in neutron-rich barium isotopes 1 2 first-author 1 1 1 corresp Corresponding author, lijx@swu.edu.cn Corresponding author, lijx@swu.edu.cn lijx@swu.edu.cn School of Physical Science and Technology, Southwest University, Chongqing 400715, China School of Physical Science and Technology, Southwest University, Chongqing 400715, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Deformed even–even nuclei Barium isotopes with quadrupole–octupole deformations are investigated on the basis of a collective model. The model describes energy levels of the yrast band with alternating parity in the neutron-rich 140,142,144,146,148 Ba. The structure of the alternating parity bands is examined by odd-even (Δ I =1) staggering diagrams. An analytical method of the collective model is proposed for the calculation of E 2 transition probabilities in alternating spectra of the nuclei 140,142,144,146 Ba. Collective Hamiltonian Quadrupole-octupole deformations Staggering effect Electric transition probability 1 Introduction 1 In atomic nuclei, the simultaneous manifestation of quadrupole and octupole degrees of freedom are correlated with typical spectroscopic characteristics of nuclear collective motion. The quadrupole mode can be applied in all regions of deducing vibrational, rotational, and transitional structures of the spectra. The display of octupole degrees of freedom is superposed in some regions. This leads to complicated shape properties and parity effects in the spectrum of the system [ 1 1 ]. It is commonly thought that the core issue of quadrupole-octupole collectivity is to resolve the breaking of reflection symmetry [ 2 2 ], this is chiefly because of the difficulty in determining the total inertia tensor of the system. Based on this situation, if simplifying assumptions that the axial symmetry is still preserved and the octupole deformations are fixed suitably with the principal axes of the quadrupole shape, both degrees of freedom are separated adiabatically. In situations like this, the collective motion can be related to the reflection asymmetric shape in reference to an octupole variable in a double-well potential [ 3 3 ], and the tunneling through the potential barrier can reasonably explain the parity shift effect observed in nuclear alternating parity bands. The above concept has been generalized for the case of simultaneously contributing quadrupole and octupole modes. The double-well potential was defined in accordance with a variable bringing contribution to not the absolute values of each deformation variables, but the different degrees of freedom [ 4 4 ]. In this way, the explicit form of the original potential according to the quadrupole and octupole deformation variables was not given. Another important issue is, if and to what extent, one may take into account a tunneling effect existed in the space of the octupole variable, β 3 , after the quadrupole coordinate, β 2 , is made to vary. In order to clarify the above question, it has proposed a collective model [ 5 5 ] for the quadrupole-octupole vibration and rotation motion of even-even nucei. The purpose of the present work is to apply this theoretical model to explain the properties of quadrupole-octupole deformations [ 6 6 - 8 8 6-8 ] in even-even nuclei: barium isotopes [ 9 9 - 12 12 9-12 ]. It can also obtain basic characteristics of energy levels, parity shift, and electric transition properties in nuclei ( 140,142,144,146,148 Ba) [ 13 13 ] with collective bands built on coherent quadrupole-octupole vibrations. 2 THEORETICAL DESCRIPTIONS 1 Now we begin with a brief review of the theoretical framework. The quadrupole-octupole Hamiltonian [ 17 17 ] for the collective motion is given by <math display="block" id="M1"><mrow><msub><mi>H</mi><mrow><mtext>qo</mtext></mrow></msub><mo>=</mo><mo>−</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>B</mi><mn>2</mn></msub></mrow></mfrac><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msubsup><mi>β</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msub><mi>B</mi><mn>3</mn></msub></mrow></mfrac><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msubsup><mi>β</mi><mn>3</mn><mn>2</mn></msubsup></mrow></mfrac><mo>+</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>β</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>3</mn></msub><mn>,</mn><mi>I</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> and the potential is <math display="block" id="M2"><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>β</mi><mn>2</mn></msub><mn>,</mn><msub><mi>β</mi><mn>3</mn></msub><mn>,</mn><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>C</mi><mn>2</mn></msub><msubsup><mi>β</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>C</mi><mn>3</mn></msub><msubsup><mi>β</mi><mn>3</mn><mn>2</mn></msubsup><mo>+</mo><mfrac><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>d</mi><mn>2</mn></msub><msubsup><mi>β</mi><mn>2</mn><mn>2</mn></msubsup><mo>+</mo><msub><mi>d</mi><mn>3</mn></msub><msubsup><mi>β</mi><mn>3</mn><mn>2</mn></msubsup></mrow></mfrac><mn>,</mn></mrow></math> with X ( I )= I ( I +1)/2. Here I is the collective angular momentum, β 2 and β 3 are the axial deformation variables, B 2 and B 3 are the mass parameters, C 2 and C 3 are the stiffness parameters, and d 2 and d 3 are the moment of inertia parameters. The last term in Eq. (2) is a coupling between quadrupole and octupole degrees of freedom. If a condition for the simultaneous present of nonzero coordinates of the potential minimum is applied, the inertial and stiffness parameters are correlated as d 2 / C 2 = d 3 / C 3 . On this occasion, the potential bottom is an ellipse that surrounds the internal potential core. If the prolate quadrupole deformation β 2 > 0 is taken into account, the motion in the octupole coordinate between positive and negative β 3 values along the ellipse surrounding the potential core. And it also uses polar variables <math id="M3"><mrow><msub><mi>β</mi><mn>2</mn></msub><mo>=</mo><mi>ρ</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>/</mo><msqrt><mrow><msub><mi>d</mi><mn>2</mn></msub><mo>/</mo><mi>d</mi></mrow></msqrt></mrow></math> , and <math id="M4"><mrow><msub><mi>β</mi><mn>3</mn></msub><mo>=</mo><mi>ρ</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>/</mo><msqrt><mrow><msub><mi>d</mi><mn>3</mn></msub><mo>/</mo><mi>d</mi></mrow></msqrt></mrow></math> , with d =( d 2 + d 3 )/2. According to the above condition, the potential energy hinges on the deformation variable, ρ , and on the angular momentum, I , and not on the angular variable, θ . The potential is expressed as <math display="block" id="M5"><mrow><msub><mi>U</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>C</mi><msup><mi>ρ</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msup><mi>ρ</mi><mn>2</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> where C is defined as 1/ C = d 2 /( dC 2 )= d 3 /( dC 3 ). Then we assume that the quadrupole and octupole modes in the collective motion have the same oscillation frequencies. Then the mass and inertia parameters have the relation 1/ B = d 2 /( dB 2 )= d 3 /( dB 3 ). The model Hamiltonian and the quadrupole-octupole oscillation wave function are obtianed and the wave function can be taken in a separable form ϕ ( ρ , θ )= ψ ( ρ ) φ ( θ ). The Schrödinger equation is separated into two equations for the variables ρ and θ <math display="block" id="M6"><mrow><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msup><mi>ρ</mi><mn>2</mn></msup></mrow></mfrac><mi>ψ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mn>1</mn><mi>ρ</mi></mfrac><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>ρ</mi></mrow></mfrac><mi>ψ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn>2</mn><mi>B</mi></mrow><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">[</mo><mi>E</mi><mo>−</mo><msub><mi>U</mi><mi>I</mi></msub><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><msup><mi>ℏ</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><mi>B</mi></mrow></mfrac><mfrac><mrow><msup><mi>k</mi><mn>2</mn></msup></mrow><mrow><msup><mi>ρ</mi><mn>2</mn></msup></mrow></mfrac><mo stretchy="false">]</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo/><mn>;</mn></mrow></math> <math display="block" id="M7"><mrow><mfrac><mrow><msup><mo>∂</mo><mn>2</mn></msup></mrow><mrow><mo>∂</mo><msup><mi>θ</mi><mn>2</mn></msup></mrow></mfrac><mi>φ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup><mi>φ</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0,</mn></mrow></math> where k is the separation quantum number. Eq. (4) with the potential (3) is similar to the equation for the Davidson potential [ 15 15 , 16 16 ], which is analytically solvable. Eq. (4) is solved analytically and get the explicit expression for the energy spectrum [ 17 17 ] <math display="block" id="M8"><mrow><msub><mi>E</mi><mrow><mi>n</mi><mn>,</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ℏ</mi><mi>ω</mi><mo stretchy="false">[</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><msqrt><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msqrt><mo stretchy="false">]</mo><mn>,</mn></mrow></math> where <math id="M9"><mrow><mi>ω</mi><mo>=</mo><msqrt><mrow><mi>C</mi><mo>/</mo><mi>B</mi></mrow></msqrt><mn>,</mn></mrow></math> n =0,1,2,..., and b =2 B /( ħ 2 d ). The eigenfunctions ψ ( ρ ) are obtained in terms of the Laguerre polynomials <math display="block" id="M10"><mrow><msubsup><mi>ψ</mi><mi>n</mi><mi>I</mi></msubsup><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mfrac><mrow><mn>2</mn><mi>Γ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mi>v</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></msqrt><msup><mrow><mo stretchy="false">(</mo><msup><mi>ρ</mi><mn>2</mn></msup><mi>a</mi><mo stretchy="false">)</mo></mrow><mi>v</mi></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>a</mi><msup><mi>ρ</mi><mn>2</mn></msup><mo>/</mo><mn>2</mn></mrow></msup><msubsup><mi>L</mi><mi>n</mi><mrow><mn>2</mn><mi>v</mi></mrow></msubsup><mo stretchy="false">(</mo><mi>a</mi><msup><mi>ρ</mi><mn>2</mn></msup><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M11"><mrow><mi>a</mi><mo>=</mo><msqrt><mrow><mi>B</mi><mi>C</mi></mrow></msqrt><mo>/</mo><mi>ℏ</mi></mrow></math> and <math id="M12"><mrow><mi>v</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msqrt><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></msqrt></mrow></math> . Under the boundary condition φ ( π /2)= φ (- π /2)=0, the Eq. (5) has two different solutions with positive and negative parities, π φ =(+) and π φ =(-), respectively <math id="M13"><mrow><msup><mi>φ</mi><mo>+</mo></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi>π</mi></mrow></msqrt><mi>cos</mi><mo stretchy="false">(</mo><mi>k</mi><mi>θ</mi><mo stretchy="false">)</mo><mn>,</mn><mi>k</mi><mo>=</mo><mo>±</mo><mn>1,</mn><mo>±</mo><mn>3,</mn><mo>±</mo><mn>5</mn><mo>⋯</mo><mo>;</mo></mrow></math> <math id="M14"><mrow><msup><mi>φ</mi><mo>−</mo></msup><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><msqrt><mrow><mn>2</mn><mo>/</mo><mi>π</mi></mrow></msqrt><mi>sin</mi><mo stretchy="false">(</mo><mi>k</mi><mi>θ</mi><mo stretchy="false">)</mo><mn>,</mn><mi>k</mi><mo>=</mo><mo>±</mo><mn>2,</mn><mo>±</mo><mn>4,</mn><mo>±</mo><mn>6</mn><mo>⋯</mo><mn>.</mn></mrow></math> The lowest states of the system in the variable θ is considered, one has k = k + =1 for φ + and k = k - =2 for φ - . With the parity dependent number, k , the Eq. (6) determines the structure of an alternating parity spectrum. The energy levers E n , k ( I ), with n =0, are equal to the yrast alternating parity sequence. With n ≠0, the levels are equal to higher energy bands, in which the rotation states are built on quadrupole-otcupole vibrations of the system. The odd-even staggering [ 18 18 - 20 20 18-20 ] is referred to as a zigzagging behavior of the nuclear inertial parameter between the odd and even angular momentum states of a rotation band. It provides some information about the fine structure of the nuclear collective spectrum in different regions of the nuclear chart. The odd-even staggering patterns can be determined as <math display="block" id="M15"><mtable columnalign="left"><mtr><mtd><mi>S</mi><mi>t</mi><mi>g</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mo>−</mo><mn>4</mn><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>+</mo><mn>9</mn><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>−</mo><mn>10</mn><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>E</mi><mo stretchy="false">(</mo><mi>I</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mn>.</mn></mtd></mtr></mtable></math> The model formalism which allows the calculation of E 2 transition probabilities for the energy spectra in the system with coherent quadrupole-octupole motion is widely applied to even-even nuclei. Then we give the formulas for calculation of E 2 transition probabilities [ 5 5 ] between levels with | n i I i ⟩ and | n f I f ⟩ <math display="block" id="M16"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2,</mn><msub><mi>I</mi><mi>i</mi></msub><mo>→</mo><msub><mi>I</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>b</mi><mn>2</mn></msub><msup><mrow><mo>〈</mo><msub><mi>I</mi><mi>i</mi></msub><mn>020</mn><mo>|</mo><msub><mi>I</mi><mi>f</mi></msub><mn>0</mn><mo>〉</mo></mrow><mn>2</mn></msup><msup><mi>S</mi><mn>2</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mn>2,</mn><msub><mi>I</mi><mi>i</mi></msub><mo>→</mo><msub><mi>I</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where b 2 is scaling constant related to effective charges, <math display="block" id="M17"><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2,</mn><msub><mi>I</mi><mi>i</mi></msub><mo>→</mo><msub><mi>I</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><mo>+</mo><mi>∞</mi></mrow></msubsup></mrow></mstyle><mi>d</mi><mi>ρ</mi><msubsup><mi>ψ</mi><mrow><msub><mi>n</mi><mi>f</mi></msub></mrow><mrow><msub><mi>I</mi><mi>f</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><msup><mi>ρ</mi><mn>2</mn></msup><msubsup><mi>ψ</mi><mrow><msub><mi>n</mi><mi>i</mi></msub></mrow><mrow><msub><mi>I</mi><mi>i</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> 3 RESULTS AND DISCUSSION 1 The descriptions of the alternating parity spectra in the nuclei 140,142,144,146,148 Ba are obtained by taking the theoretical energy levels <math id="M18"><mrow><msub><mrow><mover accent="true"><mi>E</mi><mo stretchy="true">˜</mo></mover></mrow><mrow><mn>0,</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mrow><mn>0,</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>E</mi><mrow><mn>0,</mn><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> from Eq. (6). <math id="M19"><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">[</mo><msub><mi>d</mi><mn>0</mn></msub><mo>+</mo><mi>I</mi><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></math> , where the parameter d 0 indicates the characteristic of the potential ( U ( β 2 , β 3 , I )) in the ground state. According to the respective experimental data, the parameters ω , b , and d 0 have been adjusted to the energy levels by means of a least squares minimization procedure. In the left column of Fig. 1 Fig. 1 , the obtained numerical results for the energy levels of 140,142,144,146,148 Ba are compared with experimental data. The staggering patterns illustrate that the even and odd angular momentum sequences approach each other toward higher angular momenta. However, the decrement of the staggering amplitude is not enough to provide an octupole band structure at angular momenta. For the nuclei 140,142,144,146,148 Ba, we observed this staggering effect. The respective experimental and theoretical staggering patterns are compared in the right column. In Fig. 1 Fig. 1 , it is illustrated that the experimental patterns demonstrate the predicted behaviors of alternating parity levels with increasing angular momentum in the nuclei 140,142,144,146,148 Ba. Considering in the case of transitions between states of the yrast alternating parity band, |0 I i ⟩ and |0 I f ⟩ (with n i = n f =0), the integrals in Eq. (10) have an analytic expression [ 21 21 ] <math display="block" id="M20"><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>E</mi><mn>2,</mn><msub><mi>I</mi><mi>i</mi></msub><mo>→</mo><msub><mi>I</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>a</mi><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfrac><mfrac><mrow><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>v</mi><mi>i</mi></msub><mo>+</mo><msub><mi>v</mi><mi>f</mi></msub><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo stretchy="false">)</mo></mrow><mrow><msqrt><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>v</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Γ</mi><mo stretchy="false">(</mo><mn>2</mn><msub><mi>v</mi><mi>f</mi></msub><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msqrt></mrow></mfrac><mn>,</mn></mrow></math> where <math id="M21"><mrow><mi>a</mi><mo>=</mo><msqrt><mrow><mi>B</mi><mi>C</mi></mrow></msqrt><mo>/</mo><mi>ℏ</mi></mrow></math> , <math id="M22"><mrow><msub><mi>v</mi><mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msqrt><mrow><msubsup><mi>k</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">(</mo><msub><mi>I</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow></msqrt></mrow></math> , and <math id="M23"><mrow><msub><mi>v</mi><mi>f</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>/</mo><mn>2</mn><mo stretchy="false">)</mo><msqrt><mrow><msubsup><mi>k</mi><mi>f</mi><mn>2</mn></msubsup><mo>+</mo><mi>b</mi><mi>X</mi><mo stretchy="false">(</mo><msub><mi>I</mi><mi>f</mi></msub><mo stretchy="false">)</mo></mrow></msqrt></mrow></math> . This formalism can be applied for an analysis of the electric transition rates in spectra where the collective quadrupole-octupole dynamics carry the characteristics outlined in the above cases. We calculate the E 2 reduced transition probabilities in the spectra of 140,142,144,146 Ba, where the available experimental data allow to get information about the angular momentum dependence of these quantities. Due to the scarce experimental data, the B ( E 2) value of 148 Ba was not calculated. The theoretical transition values have been determined after fitting the parameter a in Eq. (11). The scaling (effective charge) parameter, b 2 , in Eq. (9) has been set equal to 1. The specific instructions of parameter, a , and constant, b 2 , are given in the fifth part of the Ref. [ 5 5 ]. In Fig. 2 Fig. 2 , the vertical coordinate is the value of the theoretical and experimental transition probabilities ( B ( E 2; 2 + 0 + )) in the alternating parity spectra of barium isotopes. On the horizontal coordinate, the different number of neutrons signify the different barium isotopes ( 140,142,144,146 Ba). The calculated results are compared with experimental data in Fig. 2 Fig. 2 . There are also some discrepancies between theory and experiment in the E 2 transition values in 142,144,146 Ba. 4 SUMMARY 1 In summary, we have shown accurately the energy levels and the staggering patterns in the nuclei 140,142,144,146,148 Ba, as displayed in Fig. 1 Fig. 1 . From the figure we can see that the staggering patterns shows the even and odd angular momentum sequences approach each other toward higher angular momenta. It signifies a trend for the forming of an octupole band in even-even nuclei: barium isotopes. The results for E 2 transition probabilities in nuclei 140,142,144,146 Ba ( Fig. 2 Fig. 2 ) suggests further analysis of additional experimental data and tests of the formalism. The current analysis shows that, in this case, the coherent contribution of quadrupole and octupole oscillations can occur in the collective motion of nuclei. References P.A. Butler , W. Nazarewicz . Intrinsic reflection asymmetry in atomic nuclei . Rev Mod Phys . 1996 , 68 : 349 - 421 . DOI: 10.1103/RevModPhys.68.349 http://doi.org/10.1103/RevModPhys.68.349 Intrinsic reflection asymmetry in atomic nuclei R.V. Jolos , P. von Brentano . Angular momentum dependence of the parity splitting in nuclei with octupole correlations . Phys Rev C , 1994 , 49 : R2301 . DOI: 10.1103/PhysRevC.49.R2301 http://doi.org/10.1103/PhysRevC.49.R2301 Angular momentum dependence of the parity splitting in nuclei with octupole correlations H.J. Krappe , U. Wille . Collective model for pear-shaped nuclei . Nucl Phys A , 1969 , 124 : 641 - 654 . DOI: 10.1016/0375-9474(69)90656-3 http://doi.org/10.1016/0375-9474(69)90656-3 Collective model for pear-shaped nuclei V.Yu. Denisov , A.Ya. Dzyublik . Collective states of even-even and odd nuclei with β2, β4, ⋯, βN deformations . Nucl Phys A , 1995 , 589 : 17 - 57 . DOI: 10.1016/0375-9474(95)00075-C http://doi.org/10.1016/0375-9474(95)00075-C Collective states of even-even and odd nuclei with β2, β4, ⋯, βN deformations N. Minkov , P. Yotov , S. Drenska , et al .. Nuclear collective motion with a coherent coupling interaction between quadrupole and octupole modes . Phys Rev C , 2006 , 73 : 044315 . DOI: 10.1103/PhysRevC.73.044315 http://doi.org/10.1103/PhysRevC.73.044315 Nuclear collective motion with a coherent coupling interaction between quadrupole and octupole modes P.G. Bizzeti , A.M. Bizzeti-Sona . Description of nuclear octupole and quadrupole deformation close to the axial symmetry and phase transitions in the octupole mode . Phys Rev C , 2004 , 70 : 064319 . DOI: 10.1103/PhysRevC.70.064319 http://doi.org/10.1103/PhysRevC.70.064319 Description of nuclear octupole and quadrupole deformation close to the axial symmetry and phase transitions in the octupole mode P.G. Bizzeti , A.M. Bizzeti-Sona . Description of nuclear octupole and quadrupole deformation close to axial symmetry: Critical-point behavior of 224Ra and 224Th . Phys Rev C , 2008 , 77 : 024320 . DOI: 10.1103/PhysRevC.77.024320 http://doi.org/10.1103/PhysRevC.77.024320 Description of nuclear octupole and quadrupole deformation close to axial symmetry: Critical-point behavior of 224Ra and 224Th P.G. Bizzeti , A.M. Bizzeti-Sona . Description of nuclear octupole and quadrupole deformation close to axial symmetry: Octupole vibrations in the X(5) nuclei 150Nd and 152Sm . Phys Rev C , 2010 , 81 : 024320 . DOI: 10.1103/PhysRevC.81.034320 http://doi.org/10.1103/PhysRevC.81.034320 Description of nuclear octupole and quadrupole deformation close to axial symmetry: Octupole vibrations in the X(5) nuclei 150Nd and 152Sm W.R. Phillips , I. Ahmad , H. Emling , et al .. Octupole deformation in neutron-rich barium isotopes . Phys Rev Lett , 1986 , 57 : 3257 . DOI: 10.1103/PhysRevLett.57.3257 http://doi.org/10.1103/PhysRevLett.57.3257 Octupole deformation in neutron-rich barium isotopes H. Mach , W. Nazarewicz , D. Kusnezov , et al .. Influence of shell effects and stable octupole deformation on the E1 and E2 transition rates in the heavy-Ba region . Phys Rev C , 1990 , 41 : R2469(R) . DOI: 10.1103/PhysRevC.41.R2469 http://doi.org/10.1103/PhysRevC.41.R2469 Influence of shell effects and stable octupole deformation on the E1 and E2 transition rates in the heavy-Ba region S. Cwiok , W. Nazarewicz . Ground-state shapes and spectroscopic properties of Z∼58, N∼ 88 nuclei . Nucl Phys A , 1989 , 496 : 367 - 384 . DOI: 10.1016/0375-9474(89)90180-2 http://doi.org/10.1016/0375-9474(89)90180-2 Ground-state shapes and spectroscopic properties of Z∼58, N∼ 88 nuclei A. Sobiczewski , Z. Patyk , S. Cwiok , et al .. Study of the potential energy of "octupole"-deformed nuclei in a multidimensional deformation space . Nucl Phys A , 1988 , 485 : 16 - 30 . DOI: 10.1016/0375-9474(88)90519-2 http://doi.org/10.1016/0375-9474(88)90519-2 Study of the potential energy of "octupole"-deformed nuclei in a multidimensional deformation space D. Kusnezov , F. Iachello . A study of collective octupole states in barium in the interacting boson model . Phys Lett B , 1988 , 209 : 420 - 424 . DOI: 10.1016/0370-2693(88)91166-5 http://doi.org/10.1016/0370-2693(88)91166-5 A study of collective octupole states in barium in the interacting boson model N. Minkov . A model approach to quadrupole-octupole deformations in atomic nuclei . Rom Journ Phys , 2013 , 58 : 1130 - 1140 . DOI: 10.1016/RomJournPhys.58.1130-1140 http://doi.org/10.1016/RomJournPhys.58.1130-1140 A model approach to quadrupole-octupole deformations in atomic nuclei P.M. Davidson . Eigenfunctions for Calculating Electronic Vibrational Intensities . Proc R Soc Lond A , 1932 , 135 : 459 - 472 . DOI: 10.1098/rspa.1932.0045 http://doi.org/10.1098/rspa.1932.0045 Eigenfunctions for Calculating Electronic Vibrational Intensities D.J. Rowe , C. Bahri . Rotation-vibrational spectra of diatomic molecules and nuclei with Davidson interactions . J Phys A: Math Gen , 1998 , 31 : 4947 . DOI: 10.1088/0305-4470/31/21/011 http://doi.org/10.1088/0305-4470/31/21/011 Rotation-vibrational spectra of diatomic molecules and nuclei with Davidson interactions N. Minkov , S. Drenska , M. Strecker , et al .. Non-yrast nuclear spectra in a model of coherent quadrupole-octupole motion . Phys Rev C , 2012 , 85 : 034306 . DOI: 10.1103/PhysRevC.85.034306 http://doi.org/10.1103/PhysRevC.85.034306 Non-yrast nuclear spectra in a model of coherent quadrupole-octupole motion D. Bonatsos , C. Daskaloyannis , S.B. Drenska , et al .. ΔI=1 staggering in octupole bands of light actinides: "Beat" patterns . Phys Rev C , 2000 , 62 : 024301 . DOI: 10.1103/PhysRevC.62.024301 http://doi.org/10.1103/PhysRevC.62.024301 ΔI=1 staggering in octupole bands of light actinides: "Beat" patterns N. Minkov , S.B. Drenska , P.P. Raychev , et al .. "Beat" patterns for the odd-even staggering in octupole bands from a quadrupole-octupole Hamiltonian . Phys Rev C , 2001 , 63 : 044305 . DOI: 10.1103/PhysRevC.63.044305 http://doi.org/10.1103/PhysRevC.63.044305 "Beat" patterns for the odd-even staggering in octupole bands from a quadrupole-octupole Hamiltonian N. Minkov , P. Yotov , S. Drenska , et al .. Parity shift and beat staggering structure of octupole bands in a collective model for quadrupole-octupole-deformed nuclei . J Phys G: Nucl Part Phys , 2006 , 32 : 497 - 509 . DOI: 10.1088/0954-3899/32/4/008 http://doi.org/10.1088/0954-3899/32/4/008 Parity shift and beat staggering structure of octupole bands in a collective model for quadrupole-octupole-deformed nuclei N. Minkov , S. Drenska , P. Yotov , et al .. Coherent quadrupole-octupole modes and split parity-doublet spectra in odd-A nuclei . Phys Rev C , 2007 , 76 : 034324 . DOI: 10.1103/PhysRevC.76.034324 http://doi.org/10.1103/PhysRevC.76.034324 Coherent quadrupole-octupole modes and split parity-doublet spectra in odd-A nuclei http://www.nndc.bnl.gov/ensdf/ http://www.nndc.bnl.gov/ensdf/ , as of December 2015 . 10.1007/s41365-016-0128-0 This work was supported by the National Natural Science Foundation of China (Nos.11075133, U1332126, 10205019) . 2016-01-25 2016-03-02 2016-03-16 2016-09-22 2016-12-20 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 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/s41365-016-0128-0 http://dx.doi.org/10.1007/s41365-016-0128-0 . 2016 Nuclear Science and Techniques 06 27