SU(3) Symmetry in hafnium isotopes with even neutron N=100-108

Received Dec 19, 2019 Revised Feb 25, 2020 Accepted Apr 2, 2020 In this paper, we have reviewed the calculation of ground states energy level up to spin 14+, electric quadrupole moments up to spin 12+, and reduced transition probabilities of Hafnium isotopes with even neutron N = 100-108 by Interacting Boson Model (IBM-1). The calculated results are compared with previous available experimental data and found good agreement for all nuclei. Moreover, we have studied potential energy surface of those nuclei. The systematic studies of quadrupole moments, reduced transition strength, yrast level and potential energy surface of those nuclei show an important property that they are deformed and have dynamical symmetry SU(3) characters.

In this study, the calculations of energy levels of even-even 172-180 Hf isotopes have been done by using interacting boson model. The ground state band, the reduce probabilities of E2 transitions (B(E2)values), and electric quadrupole moment QL are calculated and compared with available experimental data.
Then the IBM-1 Hamiltonian in (1) can be written in general form as [18][19][20]: is the quadrupole operator ( is the quadrupole structure parameter and take the values 0 and ± √7 2 [19,20] is the octoupole (r=3) and hexadecapole (r=4) operator, and e = ed -es is the boson energy. The parameters are a0 the strength of the pairing, a1 angular momentum, a2 quadrupole, a3 octoupole and a4 hexadecapole interaction between the bosons.

Ground state band
In Figure 1 shows that the even-even 174-180 Hf isotopes have a rotational (deformed nuclei) dynamical symmetry SU(3). The rotational limit of the IBM-1 has been applied for the even-even 172-180 Hf isotopes due to the values of the 4 1 [19,20]. The best fitting for the Hamiltonian parameters are presented in Table 1 which gives the best agreement with the experimental data [21][22][23][24][25][26]. In the framework of the IBM-1, the isotopic chains of Hafnium (Z = 72) nuclei, having a number of proton-bosons holes 5, a number of neutron-bosons holes are (9, 10, 11) for even 172-176 Hf, and (10,9) for even 178-180 Hf isotopes, respectively.   Figure 2 indicates the energy of ground state band in experimental and theoretical data. This figure has shown the IBM-1 calculations for ground band (energies, spin and parity) in good agreement with those of the experimental data [22][23][24][25][26][27][28].

B(E2) and QL value
The general form of the electromagnetic transitions operator in IBM-1is [19,20,29]: Where γ0, α2 and βL (L = 0, 1, 2, 3, 4) are parameters specifying the various terms in the corresponding operators. Equation (4) yields transition operators for E2 transitions with appropriate values of the corresponding parameters. [19]: Where (s † , d † ) and (s, d) are creation and annihilation operators for s and d bosons, respectively [30], while α2and β2are two adjustable parameters that measure the strength of each term. The electric transition probabilities, B(E2) values are defined in terms of reduced matrix elements by Iachello and Arima (1987) as [20,29]: For the calculations of the absolute B(E2) values, the parameters, α2 and β2 of (4), were adjusted according to reproduce the experimental B(E2;2 1 + → 0 1 + ). Table 2 shows the values of the α2 and β2 parameters, which were obtained in the present calculations. The comparison of calculations values of B(E2) transitions with experimental data [22][23][24][25][26][27][28][29], are given in Table 3, for all isotopes under study.   Table 3 shows that, in general, most of the calculated results in IBM-1 reasonably consistent with the available experimental data, except for few cases that deviate from the experimental data. The quadrupole moment (QL) is an important property for nuclei that can determine if the nucleus is spherical (Q=0), deformed oblate (Q < 0) or prolate (Q > 0) shapes. The electric quadrupole moments of the nuclei can be derived from the transition rate B(E2,Li→Lf) values according to (6) [31]: Where L is the angular momentum. Table 4 presents the calculation of the electric quadrupole moment QL within the framework of IBM-1 for the even-even Hf nuclei. The presented results for QL are compared with previous experimental results [32].

Potential energy surface (PES)
In recent years, the potential energy surface by Skyrme mean field method was mapped onto the PES of the IBM Hamiltonian [33][34][35][36].The expectation value of the IBM-1 Hamiltonian with the coherent state (| , , 〉)is used to create the IBM energy surface [20,37]. The state | , , 〉is a product of boson creation operators ( † ) over the boson vacuum|0〉, i.e.
The energy surface, as a function of and , has been given by [10] ( , , ) = 2 (1+ 2 ) + ( +1) (1+ 2 ) 2 ( 1 4 + 2 3 cos 3 + 3 2 + 4 ) Where the αi's are related to the coefficients CL, ν2, ν0, u2 and u0 of (1). Measures the total deformation of nucleus, when = 0, the shape is spherical, and when ≠ 0 the shape is distorted. is the amount of deviation from the focus symmetry and correlates with the nucleus. If = 0 the shape is prolate, else if = 60 the shape becomes oblate. The calculated potential energy surfaces for the even-even 172-180 Hf are presented in Figure 3.From this figure all nuclei are deformed and have rotational-like characters. The prolate deformation is deeper than oblate in all nuclei.

CONCLUSION
We have reviewed theoretical calculations of 172-180 Hf isotopes with N= 100, 102, 104, 106 and 108 using IBM-1. The even-even 176-180 Hf isotopes have bosons total numbers of 14, 15, 16, 15 and 14. They were considered fully rotational (fully deformed) nuclei, and the dynamical symmetry of these isotopes is SU(3). The low-lying ground states, electric transition probabilities B(E2), and electric quadrupole moment QL are obtained for these isotopes using IBM-1 were compared with the available experimentally data. A good agreement was obtained between theoretical IBM-1for all the observable studied. The potential energy surfaces for Hf isotopes shows that all nuclei are deformed and have dynamical symmetry SU(3) characters.