Ground States Structure of Ruthenium Isotopes with Neutron N = 60, 62

In this paper, Ruthenium Isotopes with neutron N = 60, 62 have been studied the ground state bands using Matlab computer code interacting boson model (IBM-1). We apply IBM-1 formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12 + have been obtained for 104,106 Ru isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for those nuclei. The experimental and calculated R 4/2 values indicate that the even-even 104-106 Ru isotopes have O(6) dy-namic symmetry. The calculated results are compared to the experimental data and are found in good harmony with each other. The plots of the potential energy surface of both nuclei are O(6) characters.


Introduction
Recently, Ruthenium isotope has been a focus of the nuclear structure of many theoretical and experimental investigations. The low-lying even nuclei had been successfully explained nuclear collective characters using the interacting boson model-1 (IBM-1) [1]. In the first beginning the collective states can be described There is no discrepancy between neutron and proton in IBM-1. There are three dynamical symmetries indicated by U(5), SU (3) and O (6) analogous to spherical vibrator, deformed rotor, and γ-soft respectively. The microscopic a harmonic I. Hossain et al. vibrator approach (MAVA) used in investigating the lower level collective states in Ruthenium isotopes [2].
The Ruthenium isotopes have atomic number Z = 44. It belongs near to closed shell Sn (magic number Z = 50). The external forms of even 104-106 Ru isotopes have 6 9 2 g − (6 proton holes) and 10,12 9 2 g (10 and 12 neutron particles) close to magic number 50. This configuration has been investigated the ground state structure from spherical to deformed symmetry. The edifice of yrast levels and electromagnetic strength of Ru isotopes studied by many scientists [3] [4] [5] [6] [7].
The present aim particularly focuses on the structure of the ground state band and the potential energy surfaces to find the dynamical symmetry of even 104-106 Ru isotopes by the application of IBM.

Method of Calculation
The Interacting Boson Model (IBM) gives occupation to truncated model space for nuclei with N number of nucleons. It provides a quantitative description of identical particles with forming pairs of angular momentum 0 and 2.
The Hamiltonian of IBM-1 [15]: Here ε is energy of boson and V ij is the potential energy of boson between i and j.
Hamiltonian is from multi-pole form [16] ( ) ( ) ( ) ( ) ( ) H n a P P a L L a Q Q a T T a T T ε Here P is the pairing operator for s and d bosons, Q is quadrupole operator, ˆd n is number of d boson, L is operator of angular momentum, and T 3 octuplet operators and T 4 is hexadecapole operators.

I. Hossain et al.
The Hamiltonian starting with U(6) and finishing with group O(2) as given in Equation (2) is bringing to a lower state of three limits, γ-soft O(6), the vibration U(5) and the rotational SU(3) nuclei [17]. We know that in the SU(3) limits, the effective parameter is the quadrupole 2 a , in the O(6) limit the effective parameter is the pairing 0 a , in U(5) limits, the effective parameter is ε .
The Hamiltonian and eigen-values for the three limits [18]: K 1 , K 2 , K 3 , K 4 , and K 5 are other forms of strength parameters.
Then applying particular limit of symmetry (O(6), SU(3), U(5)) to determine the frame of a set of nuclei is more advantageous than full Hamiltonian of IBM-1. It comprise multi-free parameters those make it simple to fit the structure of a nuclei. A flaw chart of method of calculation is given in Figure 1

Results and Discussion
The obtained results have discussed for yrast energy level, γ-band, β-band, effective charge used to reproduce B(E2) values, transition probabilities B(E2), maxing ratio and contour plots of the potential energy surfaces using IBM-1.
The γ-unstable limit has applied for 104,106 Ru nuclei using data of experimental energy ratios (E 2 : E 4 : E 6 : E 8 = 1:2.5:4.5:6.5). In the framework of IBM-1, the even I. Hossain et al.    The best fit was taken up to 12 + of Ru isotopes with neutron N = 60, 62. The parameters were determined the experimental eigen values (E(n d , υ, L)) from the Equation (4), where n d , υ and L are quantum numbers. The parameters in the present data are shown in Table 1.
The calculated energy levels as well as experimental data are presented in Table 2. According to the weight of fitting the Ru-104 and Ru-106 nuclei are good candidates of O(6) symmetry. The calculation of γ-bands and β-bands are compared with experimental data and presented to Table 3 and Table 4. From the tables, the IBM calculations and experimental results are in good agreements [21].
The reduced electric transition probabilities give the more information on the structure of nuclei. The E2 transition operator must be a Hermitian tensor of rank two; consequently, the number of bosons must conserve.
Here T E2 is the operator of reduced matrix elements of the E2. (s † , d † ) are creation and (s, d) are annihilation operators for s and d bosons. α 2 indicated the effective quadrupole charge and β 2 is dimensionless coefficient, β 2 = χα 2 ( ) The parameters, α 2 and β 2 of Equation (6), were adjusted to reproduce the experimental ( ) . The effective charge (e B ) in present calculation is shown in Table 5. The values of e B were estimated to reproduce experimentally ( )  Table 6 for Ru isotopes with neutron N = 60, 62 in this study [21]. The calculated data of IBM-1 is good agreements with the available experimental results.
The energy surface E(N, β, γ) for O(6) limits as a function of β and γ, has been calculated [1] [22]. Here, β were indicated the total deformation of a nucleus. Figure 3 shows the contour plots in the γ-β plane resulting from E(N, β, γ) for

Conclusion
The yrast band, gamma band and beta band, electromagnetic transition and potential energy surface of 104 Ru and 106 Ru isotopes calculated in terms of O (6)