Calculation of some of the nuclear properties of even-even 172-176 Hf isotopes using IBM-1

A description of the even-even Hf isotopes interacting boson model (IBM-1). The energy levels, B(E2) transition probabilities, electric quadrupole moment Q L , and potential energy surface of those nuclei have been calculated. The calculated results are compared with the most recent experimental data. A good agreement for low lying energy states is obtained between experimental results and theoretical calculations. The contour plots of the potential energy surfaces show that the interested nuclei have rotational characters.


INTRODUCTION
The proton-neutron interaction plays an important role in quadrupole correlations in nuclei.The stimulation energies of collective quadrupole nuclei are strongly dependent on the number of nucleons outside the closed shell (Abdul & Mushtaq, 2011).The interacting boson model constitutes a simple Hamiltonian capable of describing collective nuclear properties across a wide range of nuclei, based on general algebraic group theoretical techniques, which have found applications in problems in atomic, molecular, and high energy physics (Kellman & Herrick, 1980;Van Roosmalen et al., 1982).In the interacting boson model (IBM-1), proton and neutron-boson degrees of freedom are not distinguished.It has been successful in reproducing the nuclear collective levels in terms of s and d bosons, which are essentially the collective s and d pairs of valence nucleons (Otsuka et al., 1978;Eid & Stewart, 1985;Sharrad et al., 2013).As the s and d bosons span a sixdimensional Hilbert space, the IBM-1 Hamiltonian has a group structure U (6).The three limiting symmetries O(6), SU(3) and U(5) correspond to the geometrical vibrator, respectively (Sethi et al., 1991).Hafnium isotopes with even number of neutrons N = 100, 102, 104 have been comprehensively investigated experimentally using different types of reactions.The excited states in the even 172-176   excitation reactions (Ejiri & Hagemann, 1971;Zaitz & Sheline, 1972;Bushnell et al., 1974;Khoo et al., 1976;Kondurov et al., 1981;Hague et al., 1986;Raman et al., 1987;Raymond et al., 1993;Morikawa et al., 1995).
In this study, energy levels of 172,174,176 Hf isotopes have been calculated using IBM-1.The reduced transition probabilities [B(E2)], level of positive parity state, electric quadrupole moment Q L and potential energy surface are calculated and compared with available experimental data.

March 2018
Journal of the National Science Foundation of Sri Lanka 46(1)

METHODOLOGY
In the IBM-1, the low energy collective properties of even-even nuclei are described by the interaction of s bosons (L = 0) and d bosons (L = 2).Moreover, the model assumes that the structure of the low-lying level

RESULTS AND DISCUSSION
The results for energy levels and transition energy, the transition probabilities B(E2) values and quadrupole moment Q L , and potential energy surface are discussed separately.

Energy levels and transition energy
The rotational limit of the IBM-1 has been applied for the 172-176 Hf isotopes due to the values of the ratio (Iachello & Arima, 1978;Casten & Warner, 1988).Figure 1 shows that the 174-176 Hf isotopes have a rotational (deformed nuclei) dynamical symmetry SU(3) respecting to IBM-1.The adopted Hamiltonian is expressed as follows (Iachello & Arima, 1978;Casten & Warner, 1988).
The calculated yrast band and experimental data of low-lying states for 172-176 Hf are plotted in Figure 2.

band (energies, spin and parity) are in good agreement
This Hamiltonian contains 2 terms of one-body interactions ( s and d ) and 7 terms of two-body interactions where s and d are the single-boson energies, and C L , v L and u L describe the two-boson interactions.However, it independent, as it can be seen by noting N = n s + n d .Then the IBM-1 Hamiltonian in equation ( 1) can be written in general form as follows (Arima & Iachello, 1978 is the pairing operator, is the angular momentum operator, is the quadrupole operator ( is the octoupole (r = 3) and hexadecapole (r = 4) operator, and = d s is the boson energy.
The parameters a 0 , a 1 , a 2 , a 3 and a 4 designate the strength of the pairing, angular momentum, quadrupole, octoupole and hexadecapole interaction between the bosons, respectively.is dominated by excitations among the valence particles outside the closed major shells (Eid & Stewart, 1985).The number of proton bosons N and neutron bosons N is counted from the nearest closed shells, and the resulting total boson number is a strictly conserved quantity.
The IBM-1 Hamiltonian can be expressed as follows (Arima & Iachello, 1979; Atalay & Kaan, 2005) : with those of the experimental data (Singh, 1995;Browne & Junde, 1999;Basunia, 2006), but it is deviated in the high spin (energies). 172-176Hf nuclei are shown in Tables 2 and 3, respectively.These tables show that the IBM-1 calculations are in good agreement with the experimental data for these bands.
Furthermore, we have used the root mean square deviation (RMSD) to calculate a deviation between IBM-1 energy levels and the experimental; , where m is the number of levels (Xu et al., 1989).Table 4 shows the RMSD between experimental and calculated energy levels.... (4) where 0 , 2 and L = 0, 1, 2, 3, 4) are parameters specifying the various terms in the corresponding operators.Equation ( 5) yields transition operators for E2 transitions with appropriate values of the corresponding parameters.The E2 transition operator must be a Hermitian tensor of rank two and therefore, the number of bosons must be conserved.With these constraints, there are two possible operators in the lowest order.The general E2 operator can be written as (Kassim & Sharrad, 2014): (2)   ...( 5 (2) ( ) For calculation of the absolute B(E2) values, the parameters 2 and 2 of equation ( 5) were adjusted accordingly to reproduce the experimental B(E2; ). Table 5 shows the values of 2 and 2 parameters, which were obtained in the present calculations.The comparison of calculated values of B(E2) transitions with experimental data (Singh, 1995;Browne & Junde, 1999;Basunia, 2006) are given in Table 6, for all isotopes under study.
Table 6 shows that, in general most of the calculated results in IBM-1 are reasonably consistent with the available experimental data, except for a few cases that deviate from the experimental data.
The quadrupole moment (Q L is an important property for nuclei that can determine if the nucleus is prolate (Q L > 0), deformed oblate (Q L < 0) or spherical (Q L = 0) shape.The electric quadrupole moments of the nuclei can be derived from the transition rate B(E2, L i where L is the angular momentum.Table 7 presents the calculation of the electric quadrupole moment Q L within the framework of IBM-1 for the even-even Hf nuclei.The presented results for Q L are compared with previous experimental results (Stone, 2005).

Potential energy surface (PES)
In recent years, the potential energy surface (PES) by the IBM- The energy surface, as a function of and , has been given by equation (10).
... (10) where the i L 2 0 , u 2 and u 0 of equation (1).measures the total deformation of a nucleus; when = 0, the shape is spherical, and when 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.

Figure 2 :
Figure 2: Comparison of the IBM-1 calculations with the available experimental data (Singh,

Figure 1 :
Comparison of the IBM-1 calculations with the available experimeComparison of the IBM-1 calculations with the available experimental data (Singh, 1995; Browne & Junde, 1999; Basunia, 2006) of the ratio for 172-176 Hf nuclei

Table 1 :
Adopted values for the parameters used for IBM-1 calculations.All parameters are given in MeV, except N and CHQ

Table 4 :
Root means square deviation (RMSD) between experimental and calculated energy levels (1)-176Hf nuclei Neutrons number (n)March 2018Journal of the National Science Foundation of Sri Lanka 46(1)

Table 6 :
Experimental and the IBM-1 values of B(E2) for172-176Hf nuclei (in e 2 b 2 ) † , d † ) and (s, d) are creation and annihilation operators for s and d bosons, respectively(Sharrad et al.,

Table 7 :
quadrupole moment Q L (in eb)