Investigation of Bohr–Mottelson Hamiltonian in γ-rigid version with position dependent mass

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ScienceDirect Nuclear Physics A 960 (2017) 78–89 www.elsevier.com/locate/nuclphysa

Investigation of Bohr–Mottelson Hamiltonian in γ -rigid version with position dependent mass M. Alimohammadi a,∗ , H. Hassanabadi a , S. Zare b a Physics Department, Shahrood University of Technology, Shahrood, Iran b Department of Basic Sciences, Islamic Azad University North Tehran Branch, Tehran, Iran

Received 15 December 2016; received in revised form 4 January 2017; accepted 8 January 2017 Available online 12 January 2017

Abstract In this paper, we consider the Bohr–Mottelson Hamiltonian in γ -rigid version with position dependent mass. The separation of variables has been done for the related wave equation. The obtained radial wave equation is solved for Kratzer potential. Then, the corresponding wave function, energy spectra and transition rates have been obtained for some nuclei. In addition, our results have been compared with experimental data. © 2017 Elsevier B.V. All rights reserved.

Keywords: Bohr Hamiltonian; γ -rigid version; Position dependent mass; Kratzer potential; Energy spectra; Transition rates

1. Introduction The Bohr–Mottelson collective model of nucleus [1,2], which was proposed since more than 60 years ago [3], has attracted a remarkable interest because of the possibility to derive many “new” solvable cases [4–22]. This description [1] of geometrical modes of motion has recently testified renewed interest due to the discovery of analytically solvable potentials [5–7]. Such a geometrical model treats the nucleus as a vibrating and rotating liquid drop [23].

* Corresponding author. Fax: +98 273 3335270.

E-mail address: [email protected] (M. Alimohammadi). http://dx.doi.org/10.1016/j.nuclphysa.2017.01.003 0375-9474/© 2017 Elsevier B.V. All rights reserved.

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On the other hand, the mass coefficient presented in Bohr–Mottelson Hamiltonian plays as important role as the potential energy [24]. So far, it is assumed that the mass coefficient is a constant and the same mass coefficient is used for description of the rotational motion [24]. However, one can derive information about the mass coefficient from the experimental data on excitation energies and electric quadrupole transition probabilities [24]. In the framework of the Bohr–Mottelson model and if the well-deformed axially symmetric nuclei are considered the + + + + + two “Grodzins products” E(2+ 1 )B(E2; 21 → 0g.s. ) and E(2γ )B(E2; 2γ → 0g.s. ) are inversely proportional to the corresponding mass coefficients [24]. In recent days, considerable attempts have been made for several potentials to achieve analytical solutions of Schrödinger equation, either in the usual case where the mass is assumed to be a constant or in the context of position dependent mass (PDM) [25]. The position dependent effective mass formalism has been originally introduced by von Roos in semiconductor theory [26]. Later on, this formalism has been widely used in different fields of physics such as quantum liquids [27], 3 He clusters [28], quantum wells, wires and dots [29,30], metal clusters [31], graded alloys and semiconductor hetero structures [32–38], the dependence of energy gap on magnetic field in semiconductor nano-scale quantum rings [39], the solid state problems with the Dirac equation [40] and others [41–46]. The advantage of this formalism resides in its ability to enhance the numerical calculation precision of physical observables, particularly the energy spectrum [25]. In Ref. [26], the most general non-relativistic Schrödinger equation which possesses a position dependent mass and at the same time respects hermiticity was proposed. A further step was taken in Refs. [44,45], in which a general scheme was proposed, through the introduction of a parameter (a) in the kinetic energy term, which apart from hermiticity also ensures the exact solvability of the relevant non-relativistic Schrödinger equation, in which the mass dependence on the position is reflected in the parameter (a) [47]. In this paper, we want to investigate the special form of Bohr Hamiltonian, γ -rigid version, with the Kratzer potential in the PDM formalism. γ -rigid condition means a static γ deformation which for the associated quantum Hamiltonian will have a different structure as per Pauli quantization prescription [48]. Indeed, in this version, the Bohr Hamiltonian obtains by assuming both γ and γ˙ are equal to zero. This work is organized as follows: in Section 2, the formalism of position dependent mass for the Bohr–Mottelson Hamiltonian in γ -rigid version is introduced. The separation of variables for this deformed Hamiltonian is done in Section 3 while solution for that with Kratzer potential is presented in Section 4. In Section 5, numerical results have been done for the related energy spectra and transition rates. Also, in this section we compared our results with experimental data. A conclusion of paper appears in Section 6. 2. Position-dependent mass Here a review about the basics of the formalism needed in the use of position dependent mass is presented. The main problem should be considered is the generalization of the kinetic energy term [49]. When the mass m(x) is position dependent [45], it does not commute with the momentum p = −i∇, so there are many ways to generalize the usual form of kinetic energy, P2 2m0 , where m0 is a constant mass, to obtain a Hermitian operator [49]. In order to avoid any specific choices, one can use the general two-parameter form proposed by von Roos [26], with a Hamiltonian

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H =−

 h¯ 2  δ       m (x)∇mk (x)∇mλ (x) + mλ (x)∇mk (x)∇mδ (x) + V (x) 4

(1)

where V (x) is sum of some potential and the parameters δ  , k  and λ are constrained by the condition δ  + k  + λ = −1 [49], on expressing the position-dependent mass m(x) as [44] m(x) =

B0 f 2 (x)

(2)

where B0 is a constant mass. Then by substituting Eq. (2) into Eq. (1), this equation can be written as [44] H =−

 h¯ 2  δ f (x)∇f k (x)∇f λ (x) + f λ (x)∇f k (x)∇f δ (x) + V (x) 4B0

(3)

with δ + k + λ = 2. For the special choice δ  = λ = 0 and k  = −1 or δ = λ = 0 and k = 2, equations (1) and (3) reduce to the most common Ben Daniel–Duke form [50] H =− with

 h¯ 2  f (x)∇f (x)∇ f (x) + V1 (x) 2B0

      2 h¯ 2 1 1 1 2 V1 (x) = V (x) + (1 − δ − λ)f (x)∇ f (x) + −δ − λ ∇f (x) 2B0 2 2 2

Therefore, the corresponding Schrödinger equation takes the form    h¯ 2  f (x)∇f (x)∇ f (x) + V1 (x) (x) = E(x) − 2B0

(4)

(5)

(6)

where E, h¯ and (x) are energy, reduced Planck constant and the wave function, respectively. 3. Separation of variables for the deformed γ -rigid version of Bohr–Mottelson equation with general non-central potential By attention to Eq. (6), Bohr–Mottelson equation in γ -rigid version with position dependent mass for 3-dimensional can be written as follows    h¯ 2  f (β, θ, ϕ)∇f (β, θ, ϕ)∇ f (β, θ, ϕ) + V1 (β, θ, ϕ) (β, θ, ϕ) − 2B0 = E(β, θ, ϕ) (7) where V1 (β, θ, ϕ) is non-central potential with the following form  h¯ 2 1 V1 (β, θ, ϕ) = V (β, θ, ϕ) + (1 − δ − λ)f (β, θ, ϕ)∇ 2 f (β, θ, ϕ) 2B0 2      2 1 1 −δ − λ ∇f (β, θ, ϕ) + 2 2

(8)

in which V (β, θ, ϕ) = V (β) +

f 2 (β) f 2 (β) V (θ ) + V (ϕ) β2 β 2 sin2 (θ )

(9)

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In above equation, V (β), V (θ ) and V (ϕ) are arbitrary functions [25]. On the other hand, by considering the wave function as [25], 1 R(β) (θ) (ϕ) (10) β f (β) and substituting Eqs. (8), (9) and (10) into Eq. (7) and using the standard procedure of separation of variables, one obtains the following equations [25]:     2   2B0 E − V (β) 2 − δ − λ f  (β) f  (β) d L2 + 2 − − 2− f (β) 2 β dβ 2 f 2 (β) 3β h¯      2  1 1 1 f (β) + −δ −λ − R(β) = 0 (11) 2 2 4 f (β)   2 d 2 2B0 d 2 + cot θ (12) + L − 2 − 2 V (θ ) (θ ) = 0 dθ dθ 2 h¯ sin θ  2  d 2B0 2 − V (ϕ) + 

(ϕ) = 0 (13) dϕ 2 h¯ 2 (β, θ, ϕ) =

where we have introduced the separation constants 2 and Lˆ 2 = L(L + 1), where L is the orbital angular momentum quantum number. Since we solve the Bohr–Mottelson Hamiltonian in the position dependent mass formalism for the X(3) model (γ and γ˙ are equal to zero), we use the special form of Eq. (10) in which (θ) (ϕ) = YL,M (θ, ϕ). This equation is actually obtained in a way that we consider V (ϕ) = V (θ ) = 0 in equations (12) and (13). Furthermore, it can be obviously seen that in the case of f (β) = 1 (the mass is assumed to be a constant), Eq. (11) changes to the following form  2   L2 2B0  d E − V (β) − + R(β) = 0 (14) dβ 2 3β 2 h¯ 2 This equation for R(β) = βχ(β) changes into the usual form of Bohr–Mottelson Hamiltonian in γ -rigid version  2   L2 2 d 2B0  d E − V (β) − + + χ(β) = 0 (15) dβ 2 β dβ 3β 2 h¯ 2 which is the same as Eq. (17) in Ref. [51]. 4. Solution of β-dependent equation for Kratzer potential Now, we want to present the specific form of the potential V (β) and the deformation function f (β). From the results for 3-dimensional systems reported in Ref. [44], we know that for each potential a different deformation function is appropriate [47]. In Ref. [49], the Davidson potential has been considered in the framework of the Bohr Hamiltonian with the appropriate deformation function being f (β) = 1 + aβ 2 . Also, in Ref. [47], the Kratzer potential for Bohr Hamiltonian with Position-Dependent-Mass (PDM) has been studied for which the deformation function was f (β) = 1 + aβ. In addition, in Ref. [25] the exact solutions of Schrödinger equation for a class of non-central physical potentials within the formalism of PDM have been presented. In this work, we study the γ -rigid version of Bohr–Mottelson Hamiltonian within the PDM formalism for the Kratzer potential. In this case, the appropriate form of deformation function is f (β) = 1 + aβ [47].

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We use Kratzer potential as follows V (β) = −

1 v0 + β β2

(16)

with f (β) = 1 + aβ

(17)

By substituting these equations in Eq. (11) we obtain  2   1 L(L + 1) 2 2 d 2B0 E 2 + + 2a − a β 3 dβ 2 β 2 (1 + aβ)2 h¯ 2  

2B0 L(L + 1) 2B0 v0 L(L + 1) + + 2a − 2a − β− R(β) = 0 (18) 3 3 h¯ 2 h¯ 2 where the eigenvalue can be written as  2B0 v0 2B0 L(L+1) 5 1  2  h¯ 2 (n + n + 2 + (2n + 1) 4 + 3 + h¯ 2 )a + h¯ 2 (1 + 2av0 ) 2  En,L = − 2B0 2n + 1 + 2 1 + L(L+1) + 2B0 v0 4

+

3

h¯ 2

h¯ 2 a 2 + a 2 v0 + a 8B0

(19)

and the eigenfunction has the following form λ

(τ −1, −σ a −τ −1)

ξ(β) = Nβ υ (1 + aβ)−υ−2+ a Pn

(1 + 2aβ)

(20)

where

1 L(L + 1) 2B0 v0 1 υ =− + + + 2 4 3 h¯ 2  2   a 1 L(L + 1) 2B0 v0 2B0  2 λ=a− + 2 a + a v0 − E − a + + 4 4 3 h¯ h¯ 2 1 L(L + 1) 2B0 v0 τ =1+2 + + 4 3 h¯ 2  2   a 1 L(L + 1) 2B0 v0 2B0  2 σ = −2a + 2 + 2 a + a v0 − E − a + + 4 4 3 h¯ h¯ 2

(21a)

(21b)

(21c)

(21d)

and N is the normalization constant which is calculated through N =  ∞ 0

1 [ξ(β)]2 β 2 dβ

(22)

where ξ(β) =

R(β) β(1 + aβ)

(23)

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In the limit of no dependence of mass on the deformation, i.e., a → 0, Eq. (19) has the following form 0 En,L =−

2B0 h2 ¯



(2n + 1 + 2

1 1 4

+

L(L+1) 3

+

2B0 v0 2 ) h¯ 2

(24)

which has the similar form with Eq. (19) in Ref. [51]. Also, in this limit Eq. (21) take the following form 1 L(L + 1) 2B0 v0 1 υ0 = − + (25a) + + 2 4 3 h¯ 2 2B0 0 (25b) λ = − − 2 E0 h¯ 1 L(L + 1) 2B0 v0 τ0 = 1 + 2 (25c) + + 4 3 h¯ 2 2B0 0 σ = 2 − 2 E0, (25d) h¯ so the wave function takes the form   0 0 0 ξ 0 (β) = N 0 β υ eλ β Lnτ −1 σ 0 β

(26)

which resembles Eq. (22) in Ref. [51]. It should be noted that in Ref. [51], we used the Coulomb potential instead of Kratzer potential, so obtained equations for energy and wave function in present paper exactly take the form of related ones in Ref. [51], if we choose v0 = 0 in this paper, and c = 1 in Ref. [51]. 5. Numerical results 5.1. Energy spectra To determine the energy spectra we normalize its formula in Eq. (19). For this purpose, we calculate the following ratio Rn,L =

En,L − E0,0 E0,2 − E0,0

(27)

We obtain this ratio for some levels of both n = 0 and n = 1 corresponding to the ground state and the first β band, respectively. These values for some nuclei are reported in Table 1. In this table, the free parameters “a” and “v0 ” have been chosen in a way that we achieve the best agreement with the available experimental data or the standard error have its minimum amount. This statistical quantity is defined as   N  1   2 (Rn,L )th − (Rn,L )exp i σ = (28) N −1 i=1

where N denotes the number of experimental data available for each nucleus. This value has been calculated for some nuclei and reported in the last column of Table 1. As one can see

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Table 1 Comparison of theoretical predictions (upper line) of energy spectra to experimental data [52–86] (lower line). Nuclei

R0,4

R0,6

R0,8

R0,10

R1,0

R1,2

R1,4

v0

a

σ

98 Ru

2.45 2.14 2.74 2.27 2.70 2.33 2.83 2.48 2.66 2.29 2.58 2.38 2.50 2.40 2.73 2.42 2.76 2.46 2.64 2.53 2.85 2.56 2.89 2.58 2.58 2.36 2.67 2.38 2.51 2.34 2.48 2.29 2.42 2.30 2.61 2.37 2.62 2.39 2.64 2.38 2.80 2.40 2.86 2.47 2.89 2.50 2.88 2.48 2.84 2.42

3.66 3.41 4.49 3.85 4.37 3.94 4.83 4.35 4.24 3.79 4.01 4.05 3.79 4.06 4.47 4.08 4.60 4.21 4.19 4.45 4.90 4.51 5.02 4.58 4.02 3.94 4.27 4.01 3.82 3.77 3.73 3.51 3.56 3.56 4.10 3.95 4.13 3.97 4.19 4.02 4.72 4.14 4.91 4.33 5.03 4.43 4.97 4.37 4.83 4.21

4.50 4.79 5.94 5.67 5.72 5.70 6.58 6.48 5.48 5.41 5.08 5.79 4.72 – 5.91 5.87 6.14 6.14 5.39 – 6.72 6.66 6.96 6.89 5.10 4.81 5.53 5.82 4.76 4.98 4.62 4.67 4.34 4.78 5.25 5.50 5.29 5.31 5.40 5.70 6.38 6.15 6.75 6.51 6.98 6.69 6.88 6.58 6.59 6.27

5.07 – 7.05 7.85 6.73 7.23 7.80 8.69 6.39 7.17 5.84 – 5.36 – 7.00 7.50 7.33 8.21 6.26 – 8.21 8.60 8.58 9.08 5.87 – 6.46 6.50 5.41 5.49 5.23 5.97 4.86 – 6.07 5.92 6.13 6.19 6.28 6.19 7.69 8.35 8.26 8.90 8.61 9.18 8.44 8.96 8.00 8.64

2.31 2.03 2.92 2.10 2.89 1.99 3.80 2.76 3.14 2.86 2.88 2.40 2.49 2.21 3.05 2.43 3.29 2.53 2.77 2.55 3.40 2.62 3.91 3.26 2.73 2.84 2.91 2.72 2.46 2.24 2.37 1.98 2.25 2.03 2.69 2.50 2.92 2.64 2.82 2.74 3.22 2.46 3.47 2.82 3.86 3.47 3.90 3.58 3.79 3.38

2.82 – 3.51 – 3.47 – 4.41 4.23 3.68 3.49 3.41 3.23 3.00 3.05 3.63 3.32 3.88 3.25 3.33 3.27 4.02 4.18 4.54 – 3.26 3.75 3.47 3.42 2.98 2.71 2.89 2.38 2.75 – 3.22 3.80 3.47 3.93 3.37 – 3.83 3.64 4.09 3.95 4.49 4.51 4.52 4.60 4.41 4.32

3.61 – 4.57 – 4.48 – 5.55 5.81 4.63 4.13 4.29 3.92 3.83 3.77 4.67 – 4.96 4.60 4.28 – 5.22 – 5.75 – 4.16 – 4.44 – 3.82 – 3.70 3.03 3.51 3.10 4.11 – 4.39 – 4.32 – 4.96 5.13 5.28 5.31 5.71 – 5.71 5.69 5.60 5.25

4.75

0.0100

0.33

8.72

0.0094

0.71

8.00

0.0082

0.59

11.25

0.0019

0.58

7.30

0.0010

0.47

6.17

0.0011

0.44

5.26

0.0070

0.20

8.62

0.0069

0.44

9.44

0.0048

0.60

6.89

0.0075

0.21

11.81

0.0074

0.45

13.00

0.0034

0.49

6.21

0.0053

0.31

7.41

0.0060

0.23

5.35

0.0085

0.20

5.02

0.0095

0.50

4.40

0.0098

0.32

6.60

0.0055

0.31

6.74

0.0029

0.27

7.02

0.0066

0.22

10.35

0.0077

0.52

11.96

0.0068

0.48

13.10

0.0040

0.46

12.57

0.0030

0.40

11.25

0.0036

0.44

100 Ru 102 Ru 104 Ru 102 Pd 104 Pd 106 Pd 108 Pd 110 Pd 112 Pd 114 Pd 116 Pd 106 Cd 108 Cd 110 Cd 112 Cd 114 Cd 116 Cd 118 Cd 120 Cd 118 Xe 120 Xe 122 Xe 124 Xe 126 Xe

M. Alimohammadi et al. / Nuclear Physics A 960 (2017) 78–89

85

Table 1 (continued) Nuclei

R0,4

R0,6

R0,8

R0,10

R1,0

R1,2

R1,4

v0

a

σ

128 Xe

2.77 2.33 2.62 2.25 2.34 2.16 2.14 2.04 2.87 2.53 2.70 2.43 2.50 2.32 2.15 2.28 2.82 2.32 2.92 2.56 2.60 2.38 2.33 2.31 2.29 2.33 2.81 2.49 2.60 2.35 2.39 2.33 2.65 2.35 2.41 2.35 2.57 2.19 2.59 2.23 2.72 2.31 2.93 2.56 2.88 2.53 2.84 2.49 2.85 2.48

4.59 3.92 4.13 3.63 3.36 3.16 2.90 2.52 4.95 4.46 4.39 4.16 3.78 3.66 2.91 2.70 4.79 4.08 5.13 4.55 4.07 4.01 3.33 2.91 3.23 – 4.75 4.24 4.07 3.92 3.50 3.15 4.21 3.89 3.53 3.17 3.98 3.57 4.04 3.66 4.44 3.89 5.17 4.58 4.99 4.46 4.85 4.35 4.90 4.31

6.13 5.67 5.29 5.03 4.04 – 3.35 3.54 6.82 6.70 5.75 6.03 4.70 4.69 3.36 – 6.50 6.01 7.19 6.87 5.18 – 3.99 3.94 3.84 – 6.44 6.15 5.18 5.60 4.25 4.33 5.43 5.35 4.30 – 5.03 5.07 5.13 5.23 5.84 5.69 7.28 7.01 6.91 6.71 6.64 6.47 6.72 6.38

7.31 7.60 6.13 – 4.48 – 3.63 – 8.36 9.12 6.77 6.71 5.33 – 3.64 – 7.88 8.14 8.94 9.09 5.98 – 4.42 4.49 4.22 – 7.78 8.19 5.98 5.98 4.74 4.98 6.33 6.61 4.81 – 5.78 6.68 5.90 6.89 6.90 7.64 9.08 9.70 8.50 9.18 8.08 8.57 8.21 8.62

3.45 3.57 2.96 3.35 2.27 2.77 1.92 1.93 3.80 3.30 3.16 3.24 2.54 2.91 1.91 1.93 3.79 4.27 3.56 3.75 2.61 1.95 2.14 1.87 2.06 1.83 3.33 3.04 2.60 1.87 2.23 1.89 2.76 2.66 2.26 2.54 2.67 1.79 2.83 1.98 3.15 2.70 3.80 2.46 3.72 3.01 3.50 3.11 3.85 3.78

4.03 4.52 3.50 (4.01) 2.74 2.97 2.35 2.67 4.42 4.36 3.72 3.63 3.05 3.36 2.34 2.55 4.39 (4.71) 4.21 4.80 3.16 3.90 2.62 3.35 2.53 2.77 3.94 3.88 3.15 – 2.72 2.68 3.14 – 2.76 2.99 3.21 2.70 3.36 2.71 3.72 3.54 4.45 4.17 4.35 4.20 4.12 4.07 4.47 4.55

5.10 – 4.42 (4.53) 3.43 3.16 2.90 2.78 5.61 5.16 4.73 – 3.87 3.50 2.90 2.88 5.51 – 5.48 – 4.08 – 3.30 – 3.19 3.10 5.08 5.32 4.07 – 3.46 3.36 4.28 – 3.51 – 4.09 3.72 4.25 3.74 4.74 4.49 5.73 6.38 5.56 – 5.29 – 5.63 –

9.41

0.0025

0.49

6.75

0.0022

0.42

3.81

0.0028

0.33

2.58

0.0039

0.25

12.30

0.0036

0.48

8.11

0.0035

0.21

5.21

0.0049

0.29

2.60

0.0047

0.16

10.91

0.0015

0.53

14.15

0.0081

0.44

6.41

0.0093

0.59

3.70

0.0079

0.40

3.41

0.0085

0.20

10.60

0.0065

0.36

6.41

0.0095

0.45

4.21

0.0091

0.23

7.12

0.0083

0.27

4.32

0.0084

0.30

6.03

0.0058

0.62

6.29

0.0032

0.66

8.41

0.0046

0.47

14.65

0.0062

0.73

12.75

0.0050

0.53

11.46

0.0057

0.40

11.86

0.0024

0.40

130 Xe 132 Xe 134 Xe 130 Ba 132 Ba 134 Ba 136 Ba 142 Ba 134 Ce 136 Ce 138 Ce 140 Nd 148 Nd 140 Sm 142 Sm 142 Gd 144 Gd 152 Gd 154 Dy 156 Er 186 Pt 188 Pt 190 Pt 192 Pt

(continued on next page)

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Table 1 (continued) Nuclei

R0,4

R0,6

R0,8

R0,10

R1,0

R1,2

R1,4

v0

a

σ

194 Pt

2.85 2.47 2.82 2.47 2.65 2.42 2.55 2.35

4.86 4.30 4.78 4.29 4.21 4.21 3.93 –

6.66 6.39 6.48 6.33 5.43 6.21 4.94 –

8.10 8.67 7.84 8.56 6.32 – 5.66 –

3.81 3.23 3.66 3.19 3.01 2.25 2.72 (2.38)

4.43 4.60 4.25 3.83 3.56 3.14 3.25 3.60

5.58 – 5.39 – 4.51 4.38 4.11 4.12

11.56

0.0023

0.49

10.81

0.0027

0.51

7.14

0.0028

0.54

5.82

0.0033

0.31

196 Pt 198 Pt 200 Pt

our results are in a good agreement with experimental data, especially for the 98 Ru, 106,112 Pd, 106,108,110,116,118,120 Cd, 132,134 Xe, 132,134,136 Ba, 140,148 Nd, 142 Sm, 142,144 Gd and 200 Pt nuclei. 5.2. Transition rates In addition to energy spectra, transition rates are the other important result which can be determined by expression obtained for the wave function. The reduced E2 transition probability is [51]  L20 2 2 B(E2; n1 L1 → n2 L2 ) = t 2 CL10,20 In1 L1 ;n2 L2

(29)

where ∞ In1 L1 ;n2 L2 =

βξn1 ,L1 (β)ξn2 ,L2 (β)β 2 dβ

(30)

0 L20 and CL10,20 are the Clebsch–Gordan coefficients. It should be remembered that equation (29) has been obtained with assuming that the potential is independent of θ and ϕ variables or (θ) (ϕ) = YL,M (θ, ϕ). Transition rates are obtained by using Eq. (29) and free parameters “a” and “v0 ” appeared in Table 1. The obtained values are normalized to the experimental data in a way that we obtain the minimum amount for the standard error defined as Eq. (28). Our final results for transition rates can be seen in Table 2. As one can see, there are a good agreement between our results and experimental data.

6. Conclusion In the present work, we obtain the exact solution for the γ -rigid of Bohr Hamiltonian in which the mass has been allowed to be dependent on the deformation. The energy spectra and transition rates have been calculated for some nuclei and compared the obtained results with the experimental data. Although, there are some discrepancies with experimental data for transition rates of some nuclei, especially for 114 Cd, 116 Cd, 154 Dy and 156 Er, but in the case of other transition rates and energy spectra, there is a good agreement between our results and experimental data for all nuclei presented in Tables 2 and 1, respectively.

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87

Table 2 Comparison of theoretical predictions (upper line) of transition rates to experimental data [87] (lower line). 21 → 01 is normalized to one. Nuclei

41 → 21

61 → 41

81 → 61

101 → 81

22 → 21

02 → 21

σ

98 Ru

1.04 1.44 1.35 1.45 1.36 1.50 1.28 1.18 1.56 1.56 1.36 1.36 1.23 1.63 1.35 1.47 1.30 1.71 1.60 1.78 1.50 1.54 1.35 1.68 1.75 2.02 1.29 1.99 1.25 1.70 1.01 1.11 1.01 1.16 1.02 1.47 1.04 1.34 1.25 1.47 1.06 1.36 1.04 1.40 1.74 1.84 1.22 1.62

1.12 – 1.90 – 1.99 – 1.57 – 2.26 – 1.91 – 1.68 – 1.90 2.16 1.72 – 2.64 – 2.23 – 1.95 – 3.16 – 1.89 3.83 1.70 – 1.02 0.88 1.02 1.17 1.04 0.89 1.10 1.59 1.61 1.94 1.13 1.62 1.09 0.56 3.00 2.74 1.55 2.05

1.36 – 3.11 – 3.25 – 2.08 – 3.70 – 3.09 – 2.65 – 3.06 2.99 2.46 – 4.69 – 3.58 – 3.44 – 6.54 – 3.47 2.73 2.70 – 1.05 0.49 1.05 0.96 1.09 0.44 1.16 0.63 2.15 2.39 1.25 1.55 1.18 – 5.51 – 2.29 2.27

1.83 – 5.85 – 6.01 – 3.13 – 6.73 – 5.07 – 4.64 – 5.56 – 4.36 – 9.69 – 7.28 – 7.88 – 15.52 – 7.50 – 4.52 – 1.13 0.73 1.09 0.91 1.15 – 1.33 0.29 3.06 2.74 1.50 0.93 1.31 – 11.60 – 3.57 1.86

0.97 1.62 0.66 0.64 0.63 0.62 0.67 0.63 0.43 0.46 0.60 0.61 0.80 0.98 0.61 1.43 0.65 0.98 0.41 0.43 0.53 0.64 0.72 1.09 0.42 0.50 0.79 0.71 0.73 0.63 0.99 – 0.99 – 0.97 – 0.95 0.70 0.71 1.19 0.92 – 0.95 – 0.33 0.23 0.79 –

1.04 – 1.20 0.98 1.17 0.80 0.97 0.42 1.16 – 1.09 – 1.17 0.67 1.07 1.05 1.05 0.64 1.27 – 1.25 – 1.32 – 1.82 1.69 1.31 0.88 1.14 0.02 1.00 – 1.00 – 1.00

0.76

100 Ru 102 Ru 104 Ru 102 Pd 104 Pd 106 Pd 108 Pd 110 Pd 106 Cd 108 Cd 110 Cd 112 Cd 114 Cd 116 Cd 118 Xe 120 Xe 122 Xe 124 Xe 128 Xe 130 Ba 142 Ba 152 Gd 154 Dy

0.17 0.28 0.40 0.03 0.01 0.47 0.43 0.47 0.27 0.12 0.50 0.22 1.12 0.85 0.41 0.17 0.57

0.99 0.66 – 1.01 0.37 – 0.99 0.50 – 1.00 0.64 – 1.56 0.55 2.47 1.21 1.06 – (continued on next page)

88

M. Alimohammadi et al. / Nuclear Physics A 960 (2017) 78–89

Table 2 (continued) Nuclei

41 → 21

61 → 41

81 → 61

101 → 81

22 → 21

02 → 21

σ

156 Er

1.06 1.78 1.04 1.56 1.03 1.73 1.24 1.48 1.21 1.19

1.14 1.89 1.09 1.23 1.07 1.36 1.54 1.80 1.56 1.78

1.31 0.76 1.19 – 1.12 1.02 2.01 1.92 2.20 –

1.60 0.88 1.32 – 1.24 0.69 3.00 – 3.58 –

0.94 – 0.95 1.91 0.96 1.81 0.73 – 0.77 1.16

1.02 – 0.99 – 0.99 0.01 1.02 0.07 1.04 –

0.80

192 Pt 194 Pt 196 Pt 198 Pt

0.78 0.72 0.59 0.29

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