Bohr model description of the critical point for the first order shape phase transition

Accepted Manuscript Bohr Model description of the critical point for the first order shape phase transition

R. Budaca, P. Buganu, A.I. Budaca

PII: DOI: Reference:

S0370-2693(17)30911-5 https://doi.org/10.1016/j.physletb.2017.11.019 PLB 33330

To appear in:

Physics Letters B

Received date: Revised date: Accepted date:

22 June 2017 26 September 2017 9 November 2017

Please cite this article in press as: R. Budaca et al., Bohr Model description of the critical point for the first order shape phase transition, Phys. Lett. B (2017), https://doi.org/10.1016/j.physletb.2017.11.019

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Bohr Model description of the critical point for the first order shape phase transition R. Budacaa,∗, P. Buganua , A. I. Budacaa a Department

of Theoretical Physics, Horia Hulubei National Institute for Physics and Nuclear Engineering, Reactorului 30, RO-077125, POB-MG6, Bucharest Magurele, Romania

Abstract The critical point of the shape phase transition between spherical and axially deformed nuclei is described by a collective Bohr Hamiltonian with a sextic potential having simultaneous spherical and deformed minima of the same depth. The particular choice of the potential as well as the scaled and decoupled nature of the total Hamiltonian leads to a model with a single free parameter connected to the height of the barrier which separates the two minima. The solutions are found through the diagonalization in a basis of Bessel functions. The basis is optimised for each value of the free parameter by means of a boundary deformation which assures the convergence of the solutions for a fixed basis dimension. Analyzing the spectral properties of the model, as a function of the barrier height, revealed instances with shape coexisting features which are considered for detailed numerical applications. Keywords: Collective states, Shape phase transition, Critical point, Shape coexistence, Sextic potential. PACS: 21.60.Ev, 21.10.Re, 27.70.+q 1. Introduction The Bohr-Mottelson (BM) model [1, 2] in its extended version [3, 4, 5] represents the fundamental phenomenological frame for the description of collective behavior in atomic nuclei [6, 7]. The model’s shape variables β and γ are nowadays geometrical concepts which are universally implemented where the nuclear shape is considered. The information on the actual shape of the nucleus is contained in the collective potential energy which depends in a mixed way on both β and γ. Special exactly solvable instances of the model offer few useful references in what concerns the dynamical conditions, such as the spherical vibrator [1], axially symmetric [2] and asymmetric [8] rotors. Each of these solutions, traditionally referred to as shape phases, have specific spectral characteristics emerging as a consequence of their association to the dynamical symmetries U (5) [9], SU (3) [10] and O(6) [11] of the Interacting Boson Model (IBM) [12]. There are situations when the nucleus cannot be certainly categorized, exhibiting properties associated to different shape phases. Such nuclei are considered as critical points for a shape phase transition (SPT) undergoing along the variation of the nucleon numbers. The understanding of such transitional nuclei has considerably improved since the introduction of BM formulations with an infinite square well (ISW) potential. This is how the critical point solutions E(5) [13] and X(5) [14], associated with the transition from the spherical vibrator to the asymmetric and respectively symmetric rotors, came to existence. Although, analytically the two models are fairly ∗ Corresponding

author Email address: [email protected] (R. Budaca)

Preprint submitted to Physics Letters B

similar, conceptually their ISW β potentials approximate quite different phenomenological pictures [15, 16]. In case of E(5), the potential is just flat, marking the point of the transition between increasingly soft spherical nucleus and a deformed one. Whereas, in the X(5) situation, the critical point potential has two degenerate minima, one spherical and another one deformed, separated by a small barrier. The realization of spherical or deformed shapes is then achieved by having one of the minima deeper than the other. In this context, it is said that E(5) solution corresponds to the critical point for a second order SPT, while X(5) for a first order. The description of a second order SPT from one nucleus to another can be studied analytically within the Algebraic Collective Model (ACM) [17, 18, 19, 20] by employing a quartic potential, or using the quasi-exactly solvable models with a sextic potential [21, 22, 23, 24, 25]. Solving the problem for a potential corresponding to a SPT of the first order is a much harder task which cannot be achieved with neither of the mentioned approaches. The ACM diagonalization procedure based on pseudo-harmonic oscillator functions is excellent for solving Bohr Hamiltonians for a very large range of potentials which are however restrained to have a single minimum, be it flat or sharp. While, the sextic potential with two minima is quasi-exactly solvable only in special situations with limited physical relevance. In this letter we propose a procedure to diagonalize the Bohr Hamiltonian for potentials with multiple minima. For a qualitative impact, one will focus here only on the sextic potential with two degenerated minima specific to a critical point of a first order SPT. Such a potential energy is supposed to describe two coexisting shapes. Presently, November 13, 2017

shape coexistence is usually approached with microscopic models [26] including IBM based studies with configuration mixing [27]. Nevertheless, the information on the nuclear shape is extracted by defining a collective wave function or respectively mapping the energy in β and γ coordinates. Therefore, a purely geometric description of this phenomenon is of a major interest. This is achieved here by investigating in what conditions a critical potential with double degenerated minima exhibits shape coexistence features. This connection between criticality and shape coexistence was investigated in the past within IBM with configuration mixing [28]. The choice for the diagonalization basis is inspired by the algebraic structure of the critical point solutions with ISW, which were found to be closely related to the Bessel differential equation and implicitly the Euclidean dynamical symmetry [29, 30, 31]. Indeed, E(5) is an exact, while X(5) is a partial realization of the five-dimensional Euclidean symmetry Eu(5). Euclidian dynamical symmetry is partially realized to a different extent also in other BM solutions with ISW, such as X(3) [32], X(4) [31, 33], Z(4) [34] and Z(5) [35]. Obviously, Euclidean dynamical symmetry is of central role for the description of SPT phenomena [36, 37]. As a matter of fact the Eu(5) also emerges at the triple point of the shape phase diagram of the IBM [38]. It is then natural to assume that a suitable diagonalization basis for the critical point of a first order SPT, must have an algebraic structure incorporating the Euclidean dynamical symmetry.

which is the lowest order polynomial allowing multiple minima. Its associated eigen-energy is scaled as follows β (a, b, c) = a1/2 β (1, ba−3/2 , ca−2 ).

Due to this scaling relation one must solve only the eigenvalue equation for potentials of the type: v(β) = β 2 + μβ 4 + νβ 6 .

(5)

The above potential has two minima only if μ < 0 and ν > 0, and which are localized at βs = 0 and  μ2 − 3ν − μ , (6) βd = 3ν where indices s and d denote the spherical and deformed minima. One will concentrate further only on potentials with both minima degenerated, i.e. at the same energy. In this case the potentials to be considered are expressed as a function of a single parameter: v(β) = β 2 − 2qβ 4 + q 2 β 6 .

(7)

The minimum value v = 0 for this potential is achieved √ for βs = 0 and βd = 1/ q. In order to have a consistent treatment for the whole range of considered potentials, one √ will make a final change of variable y = qβ. This induce a y-form for the β part equation (2)   ∂2 L(L + 1) 4 ∂ − 2− + + v(y) Ψ(y) = EΨ(y), (8) ∂y y ∂y 3y 2

2. Theoretical framework

with the potential

Our aim is the study of the critical point for a first order SPT between spherical and axially deformed nuclei by means of the Bohr Hamiltonian:  1 ∂ ∂ 2 1 ∂ 4 ∂ β + 2 sin 3γ H = − 2B β 4 ∂β ∂β β sin 3γ ∂γ ∂γ  3 Q2k 1    + V (β, γ), (1) − 2 4β sin2 γ − 23 πk k=1

v(y) =

 1  2 y − 2y 4 + y 6 , 2 q

(9)

and an energy E = β /q. Such a form of the potential, indeed provides an invariant position for the deformed minimum and the barrier peak, as can be seen in Figure 1. The minimum is therefore fixed at yd = 1, while the maximum at y = 1/3 whose height is [8/(27q)]2 . A highly effective method for solving the Schr¨odinger equation for general polynomial potentials is based on the expansion of the wave function into a Fourier-Bessel series [39, 40, 41]. Basically, truncating the usual asymptotic boundary condition of the y wave function

where B is the mass parameter, while Qk (k = 1, 2, 3) denote the three projections of the angular momentum on the principal axes of the intrinsic frame of reference. Under the hypothesis of a separable potential of the form V (β, γ) = [v(β) + u(γ)]2 /2B, one will consider an adiabatic decoupling of the β and γ shape fluctuations as in case of the well known X(5) model [14]. Basically, employing a small angle approximation and after performing the integration over the Euler angles, one obtains the following radial-like differential equation for the β shape variable   L(L + 1) 1 ∂ 4 ∂ β + + v(β) Ψ(β) = β Ψ(β), (2) − 4 β ∂β ∂β 3β 2

lim Ψ(y) = 0,

y→∞

(10)

as Ψ(yW ) = 0 with yW being an appropriately chosen finite limit, facilitates the use of Bessel functions of the first kind as a basis for precise diagonalization of the y Schr¨ odinger equation (8) for any polynomial potential. To define the basis functions, one must first solve the Schr¨ odinger equation for the ISW potential

0, y ≤ yW , v˜(y) = (11) ∞, y > yW ,

One considers now a general sextic potential v(β) = aβ 2 + bβ 4 + cβ 6 ,

(4)

(3) 2

9

2.0 q  0.1

8

0.2 0.5

1.5

7

1

6

2 1.0

yW

vy

5 10

5 4 3

0.5

2 1

0.0 0.0

0.5

1.0

1.5

0 -1 -1,5 q =0.0316

2.0

y

defined in terms of the limiting value yW which gives the position of the infinite wall. This leads to the following differential equation:   L(L + 1) ˜ ∂2 4 ∂ ˜ + − 2− Ψ(y) = λΨ(y), (12) ∂y y ∂y 3y 2

1

1,5

2 q =100

3. Numerical results The boundary limit yW is understood here as a nonlinear optimization parameter for the diagonalization basis. It is determined by matching the diagonalization results with the original unbounded problem defined by yW → ∞ with any prescribed precision ε. From the computational point of view, this condition can be rewritten as   ELn (yW ) − ELn (yW + δ) < ε, (18) β β

(14)

while αn = yW λ are associated zeros indexed by the order n = 1, 2, 3.... Having now a set of orthogonal functions, the solution of Eq.(8) can be expressed as an eigenfunction expansion: n max ˜ νn (y), ΨLk (y) = Akn Ψ (15) n

where δ is a large quantity. Besides its dependence on the potential v(y) and the accuracy ε, the boundary limit also depends on the quantum numbers [40, 41]. Nevertheless, if one take into consideration a limited portion of the full spectrum, one can accommodate the same convergence accuracy for all chosen states with a fixed yW value. An optimal boundary parameter for a fixed set of quantities v(y), L, nβ and ε is distinguished by the minimal size of the associated truncation of the basis [39]. For the purpose of our particular problem, we have chosen a sufficiently large size for the truncated basis, nmax = 20, in order to have convergent eigenvalues for all considered states. The boundary limit associated to a chosen potential v(y) is then fixed in such a way as to have a convergence accuracy ε = 10−7 for the highest considered energy state of Eq.(8), which is set to be (L = 20, nβ = 1).

where nmax → ∞, while k designate different eigensolutions. One can always truncate the eigenvalue space to a finite dimension nmax which assures the desirable convergence to the exact eigenvalues. In this case, the eigenvalues are determined by diagonalizing the Hamiltonian matrix defined in the nmax × nmax truncated space of the eigenfunctions (13) as 3 2i (ν,i) 2 vi (yW ) Inm αn , δnm + 2 i=1 yW q Jν+1 (αn )Jν+1 (αm )

0,5 log10 q

can be consistently expedited by employing some recurrence relations [40, 41]. The set of final eigenvalues for a certain value of ν and implicitly a fixed angular momentum L belong to different β vibrational bands. The lowest eigenvalues correspond to the ground band, the second lowest to the first excited β band and so on. Therefore, k = nβ + 1, where k is the order of the eigensolution. The diagonalization procedure also provides the coefficients of the expansion (15) needed for the calculus of the electromagnetic transitions and other relevant observables.

where λ is a constant. The normalized solutions of the above equation are √ −3 2y 2 Jν (αn y/yW ) ˜ Ψνn (y) = , (13) yW Jν+1 (αn )

Hnm =

0

Figure 2: The boundary value yW as function of the potential parameter q determined for a basis with nmax = 20 and with a convergence precision ε < 10−7 .

Figure 1: Potential v(y) given in arbitrary units as a function of the √ variable y = qβ for selected values of q.

where Jν are Bessel functions of the first kind with  9 L(L + 1) + , ν= 4 3

-0,5

(16)

where v1 = v3 = 1 and v2 = −2. The computing of the integrals

1 (ν,i) = x2i+1 Jν (αn x)Jν (αm x)dx, x = y/yW , (17) Inm 0

3

20

A more deeper insight into the phenomenology of the collective conditions associated to such a model is achieved by studying the probability density distribution

10β+ 14g+

 2 ρLnβ (y) = ΨLnβ (y) y 4

16 8β+

β

(ELn - E00 )/(E20 - E00 )

12g+

12

in respect to the y variable and the integration metric dy. The evolution of this quantity as function of the scaled variable y as well as the parameter q is depicted in Figures 4(a) and 4(b) for the ground state and the excited 0+ β state, respectively. The ground state probability starts at small q values with a very sharp and symmetrical peak centered in the deformed minimum of the potential. This situation when most of the low lying states are below the energy of the separating barrier is entirely consistent with the SU (3) prescription. It is interesting the fact that despite the presence of two minima, the ground state and its rotational excitations prefer the deformed minimum. This predisposition is actually due to the centrifugal stretching. In contradistinction, the vibrational states prefer the spherical minimum when q → 0. The predilection of vibrationallike states for the spherical region, and the persistence of regular SU (3)-like rotational bands up to high energies and angular momenta in the deformed region was also evidenced in the comprehensive quantum analysis of high barrier IBM Hamiltonians made in Ref.[44]. The maximal separation of the rotational and vibrational equilibrium deformations is actually the reason why the vibrational states in rotational nuclei are going to infinity. Increasing further the q value, the ground state probability distribution becomes asymmetric (Figure 4(e)) by ”spilling” some of the probability into the spherical minimum. This change happens when the ground state is in vicinity of the separating barrier (Figure 4(c)). It can be decomposed in two closely positioned peaks, one corresponding to the spherical well, and the other to the deformed minimum. Such a picture would be consistent with the shape coexistence usually associated to a SPT of the first kind. The profile of ρ0,1 (y) for this critical case supports this statement. It presents the usual two peak structure corresponding to the two turning points of the β vibration around the equilibrium ground state deformation, but the more prominent peak corresponds to the smaller deformation turning point. This relative peak height distribution is completely reversed from the case of pseudoharmonic β oscillations within potentials along a second order SPT [19, 20]. This effect subsides with increasing q. Indeed, the ground state probability distribution becomes more extended over both minima as the barrier is finally ignored. Correspondingly, the two β vibrational peaks acquire similar heights. A similar critical value qcβ can be identified in the evolution of the probability distribution for the 0+ β state. The energetic picture for this case is shown in Figure 4(d). This critical value marks the change from the vibration inside the spherically deformed well (q < qcβ ) to the usual vibration between the two minima (q > qcβ ). The redistribution

6β+ 10g+ 4β+ 8g+

8

2β+ 6g+ 0β+

4

4g+ 2g+

0 -1,5

-1

-0,5

0

0,5 log10q

1

1,5

(19)

2

Figure 3: The low-lying energy spectrum of the ground band and first excited β band as a function of log10 q.

The convergence of lower states is then confirmed through numerical try-outs. The boundary values found in this way are interpolated into a graphical representation in Figure 2 as a function of q. The increase of the yW values with q is a logical consequence judging by the evolution of the potential v(y) as function of q from Figure 1. Using the yW values from Figure 2 we are now able to study the behaviour of the low lying energy spectrum as a function of q and implicitly the change of the barrier height. This is done in Figure 3, from where one can see that our model recovers in the regime of large q values the predictions of the X(5)-β 6 solution [42]. This is to be expected, since in the large q limit, the potential (7) reduces only to the dominant sextic term. In the other limit, that of very small values of parameter q, the ground band states arrange themselves exactly in a rotational L(L + 1) pattern, while the β excited states go to infinity. These spectral characteristics correspond to a rigid nuclear shape with an axial symmetry. Given the fact that the X(5)-β 6 solution is a fairly close facsimile for the X(5) critical point symmetry [42], the present model can be considered as a Bohr model realization of the half-side from the IBM symmetry triangle [15] which connects the SU (3) dynamical symmetry with the X(5) critical point. The fact that one recovers these two limits, supports the correctness of the employed diagonalization procedure. In the regime of intermediate values q < 1, there are situations with degenerate ground and β states, including the spectrum with + 6+ g and 0β degeneracy, which was suggested as a possible signature for nuclei with critical behavior in Ref.[43]. 4

0.5

0.5

(a) 0.0

log10 q

log10 q

0.0

-0.5

-1.0

-0.5

-1.0

-1.5

-1.5 0.0

0.5

1.0

1.5

2.0

0.0

0.5

1.0

y 120 100

100

60

2g

2g

40

0g

0g

20

0Β

40 20 0.5

1.0

vy

vy

0Β

d

80

60

0 0.0

0 0.0

1.5

0.2

0.4

0.6

y

0.8

1.0

1.2

1.4

y

4

4

e

qcg  0.0798 yW  1.884 0Β

f

2 1

qcΒ  0.0366 yW  1.545

0g

3

0g

ΡLn Β y

3 ΡLn Β y

2.0

120

c

80

0 0.0

1.5

y

0Β

2 1

0.5

1.0

0 0.0

1.5

y

0.2

0.4

0.6

0.8

1.0

1.2

1.4

y

Figure 4: Contour plots of the density of probability distribution as a function of y and q for ground state (a) and for the first excited β state + (b). Consecutive contour lines are separated by 0.2 in arbitrary units in both cases. The potential and the absolute values for the 0+ g , 2g g β and 0+ energy levels are given in the same arbitrary units for the critical values q (c) and q (d). The critical values are pointed out in the c c β corresponding contour plots with a white horizontal line. The probability distribution y-dependent profiles for the ground state and the β band head state are shown in figures (e) and (f).

of the probability density in the vibrational regime at the critical value qcβ shown in Figure 4(f) suggests a triple node vibration which is another shape coexisting feature.

ity of experimental data concerning especially the β band states. The comparison of the theory and experiment is made in Figure 5. Transition rates are calculated using q values fixed from the energy fits and following the procedure from Ref.[49]. It amounts to the calculation of a y matrix element accompanied by a Clebsch-Gordan coefficient. As can be easily observed, the ground band states as well as associated transition rates are extremely well reproduced. The major discrepancies are coming from the β band states. This is a well known issue of the geometrical description of critical point nuclei. It can be managed by an adjustable slope of the outer wall [50], which would imply higher power terms in β. Nevertheless, the energy range for the β band in general is fitted quiet well and the

The model is experimentally realized in quite a few nuclei. The critical point candidates around the N = 90 rare earth region, as well as some Pt and Os isotopes are some off-hand suspects [33]. The rotational nuclei from the transuranic region are other feasible applications of the model in its small q regime. But, as the most interesting phenomena happen in the vicinity of the two critical parameters, qcg and qcβ , we will present here some nuclei whose fitted value of q matches this interval of values. Thus, the 238 Pu, 152 Nd and 170 Hf nuclei were selected due to their reasonable fit results as well as considering the availabil5

16+

16+

40




197

6+

168 (10)

101 14+

4+ 192 12+

12




67

+

31

1.1 (4) 1.4 (4)

10

8

4

+

5

4+

4+

2+

+

0+

2 0+

g Exp

238

Pu

104

4

168 (10)

4+

+

16+

210+65 -41

254

14+

14+

+

2 10+ 0+ 188

8+

2+

1.5

0+

8+

6+ 126 (26) 4+

144

2+ 0+

g

12 2.7

2 0+

g

Th.

152

Exp

Nd

10

204+15 -13




189 (9) 6+ 4+

146

+

144 (5)

2+ 0+

g




g

Th.

Exp

4+

+

4 2+ 8 + 0+ 200 6+ 180 4+ 2+ 0+ 170

95

216

+

6+ 126

230

169 (8)

4+ 131 (10)

207+29 -23

8+

1.3

6+ 166

243 12+

10+

39 7.8

205+35 -27 12+

73

197

178

6+ 161

6+

12+

0.8

172 6+

12+

10+

180




205




10+

+

14+

+

0+ 10+

16+

14+ 162 (9)

2

2+ 0+ 186

20

0

212

14+

30

8

16+

16+

57 20 2 152

2+ 0+

36

5.5

g

Hf

Th.

Figure 5: Theoretical results for ground and β band energies normalized to the energy of the first excited state 2+ g and associated E2 transition + 238 Pu [45], 152 Nd [46] probabilities similarly normalized to the B(E2, 2+ g → 0g ) value, are compared with the available experimental data for and 170 Hf [47, 48] nuclei. Theoretical predictions of the fits against the free parameter q whose resulting values are 0.0422, 0.0535, 0.0833 correspond to boundary parameters yW = 1.607, 1.710, 1.928 and rms deviations 0.93, 1.09 and 1.21, respectively.

monopole transition strength is given as [52]

15


0+ |2 |0+g 2

ρ2if (E0) =



3 4π

2

4 Z 2 βM f |β 2 |i2 ,

(20)

10

where Z is the charge number and βM is the scaling factor of the BM. The model dependent contribution in the above equation is f |β 2 |i2 = Ψf (y)|y 2 /q|Ψi (y)2 , which is plot+ ted for the transition 0+ β → 0g in Figure 6 as a function of q. It has a maximum value in between qcg and qcβ at q = 0.0589, and becomes essentially null at very small barriers. This picture is consistent with the strong mixing between spherical and deformed states [51] which happens in the interval [qcβ , qcg ]. Although there are very few measurements concerning the E0 transitions [53], there is an experimen+ −3 for 238 Pu tal value of ρ2Exp (E0; 0+ β → 0g ) = 180(110)·10 [51]. To calculate the theoretical value we first obtained βM by matching the theoretical expression for the average ground state quadrupole deformation and the tabulated value 0.288 [54] extracted from the analysis of B(E2) transition rates. Then using the corresponding fitted parame+ −3 ter q = 0.0422 one obtained ρ2T h (E0; 0+ β → 0g ) = 81 · 10 which is in good agreement with the experimental value.

5

0 -1.5

-1.0

0.0

-0.5

0.5

1.0

log10 q Figure 6: Monopole transition probability in units of as a function of log10 q.



 3 2 4π

4 Z 2 βM

predictions for the inter-band transitions are in a qualitative agreement with the few available experimental values. A telling observable in what concerns the shape coexistence is the strength of the monopole transition between low lying 0+ states [26, 51]. More precisely, it is stronger in nuclei with large variations of nuclear radius associated with different 0+ configurations. This variation is in turn related to the difference between the quadrupole deformation of the two states. In the geometrical model, the

4. Discussion We have used here the Bessel-Fourier expansion method to construct a basis for the diagonalization of a prolate version of a Bohr Hamiltonian with a sextic potential in the 6

β shape variable. The potential is restricted to have two degenerated minima, spherical and deformed. This condition renders the model to a single free parameter when only the ground and β band states are considered. The same parameter, and implicitly the shape of the potential, dictates the boundary limit yW used to define the Bessel functions of the basis. yW plays a similar role as the parameter β0 in the ACM model [18, 20] which optimizes the diagonalization basis of pseudo-harmonic functions associated to a Davidson potential with a minimum in β0 . The calculated spectrum as function of the free parameter recovers the SU (3) and X(5)-β 6 predictions in the small values and asymptotic limits of the free parameter. Depending on the height of the barrier relatively to the low lying energy states, one distinguishes cases of coexisting shapes in the ground state and 0+ β excited state. Therefore, not all potentials having degenerated minima exhibit by default shape coexistence. The emergence of coexisting shapes, depends on the energetic rapport between the ground or excited states and the potential barrier. The critical situations of shape coexistence are discussed from a phenomenological point of view using the density of probability distribution for the deformation and the E0 transition. The physical relevance of these critical phenomena is supported by few experimental realizations of the model. Similar critical potentials with shape coexisting features are expected to arise also for higher excited states. In conclusion, the expansion of the basis states in Bessel functions, which are closely related to the Euclidean dynamical symmetry, is shown to be very successful in treating the critical phenomena implying multiple deformation minima. The perspectives of the procedure presented here are numerous, starting from considering an actual SPT with minima of different depths and continuing with the introduction of γ-softness. Finally, the method could be applied to a general potential with β and γ mixed terms, fact which would greatly expand the mapping of the collective behaviour throughout the nuclide chart.

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43]

Acknowledgments [44] [45] [46] [47] [48] [49] [50] [51]

This work was supported by CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2016-0092. 5. Bibliography [1] A. Bohr, Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 26 (1952) 14. [2] A. Bohr, B.R. Mottelson, Mat. Fys. Medd. Dan. Vidensk. Selsk. 27 (1953) 16. [3] A. Bohr, B.R. Mottelson, Nuclear Structure, Vol. 2, Benjamin, Reading, Massachusetts, 1975. [4] D.J. Rowe, Nuclear Collective Motion: Models and Theory, Methuen, London, 1970. [5] J.M. Eisenberg, W. Greiner, Nuclear Theory Vol. I: Nuclear Models, North-Holland, Amsterdam, 1975. [6] L. Fortunato, Eur. Phys. J. A 26 s01 (2005) 1. [7] P. Buganu, L. Fortunato, J. Phys. G: Nucl. Part. Phys. 43 (2016) 093003.

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7

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