Mixed symmetry states for 70Zn, 72Ge, 74Se, 76Kr isotones by using IBM-2

In present study we have calculated some nuclear properties for 70Zn, 72Ge, 74Se, 76Kr isotones, by using interacting boson proton and neutron model and calculated some properties for these isotones, firstly: excited energy levels by guessing a small number of parameters in Hamiltonian operators, secondly: display the energy levels (25+,31+,51+,11+) that have Mixed symmetry of state (MSS) property which gets a quick response when changing majorana term ζ2, as a result of mixing between the wave function of protons and wave function of neutrons. Thirdly: it has been calculated energy ratios 41+/21+ and 61+/21+ it was found that these isotones have a varied dynamic symmetries. Starting from the symmetry U(5) of the 70Zn nuclei then passing through the transition region between (U5-O6) for 72Ge, 74Se nuclei and finally approaching to the O(6) symmetry for 76Kr nuclei. As well as have been calculated the probability for electric quadruple transition B(E2), magnetic dipole B(M1) and mixing ratios δ(E2/M1), which All express a good Agreement compared with experimental data.


Introduction
The interacting boson model (IBM) is suitable for describing intermediate and heavy atomic nuclei. In the IBA-model the collective properties of a nucleus are described in terms of a system of bosons which can be either in an s (l=0) or in a d (l=2) state. In the formation of these collective pairs only the valence nucleons will be important. This means that the number of bosons is equal to the number of pairs of particles outside the closed shell [1]. Those low lying of the eager energy levels in some nuclei can be obtained by adjusting a small number of parameters in Hamiltonian operators and it is possible to solve this operator exactly for certain sets of parameters using group theoretical methods. The Hamiltonian can be regarded as a general rotation in a six dimensional space. The six dimensions are formed by the s-boson and the five components of the d-boson, d2, d1, d0 d-1, d-2. This means that the general Hamiltonian can be discussed in terms of the group U (6), of all unitary transformations in six dimensions [2].

The Interacting Proton and Neutron Boson Model
The Hamiltonian in IBM-2 is define as: [3] H = ε d (n dν + n dπ ) + κ(Q ν . Q π ) + V νν + V ππ + M νπ (1) Where εd is the energy of dbosons, nρ is the number of d bosons, where ρ correspond to π (proton) or ν (neutron) bosons, Q ν and Q π is the quadrupole -quadrupole interaction between proton and  Starting with table 1: it might make watched that those parameter of bosons energy (ε) will be diminishing over worth when proton numbers try dependent upon mid shell (N=39) et cetera it increments again, this implies that those bosons energy is proportional of the energy level particularly for those state 2 1 + by the bosons in the mid-shell need least energies Since they need aid under those impact of the two shells (28-50). The (κ) parameter declines proportionally with the increment of the protons amount contingent upon the collaboration quality of the electric quadruple to protons What's more neutrons. Those (χπ) will be the deformity parameter for protons and the qualities would continuously expanding linearly with the expanding protons numbers. The qualities for majorana parameters terms (ζ1=3, ζ2) would chose for certain qualities Also are movable with the energy about mixed-symmetry states (MSS) for some energized vitality levels. The best fit to the excited energy levels has been observed specially for the ground band (i.e. the states 2 1 + , 4 1 + , and 6 1 + ), while the second and third bands are found to show acceptable agreement. These bands have some properties of the β and γ band so they are called (quasi beta and gamma band). All of the momentum and parity for some energy levels that had not been previously identified experimentally for each isotopes are identified throughout this study. Figure 2. hint at a correlation the middle of test vitality levels [7,8,9,10] and ascertained values for at isotones contemplated.

Mixed symmetry state:
Studying the influence of Majorana parameters (ζ1,3 , ζ2 ) on the excitation energy level for 70 Zn, 72 Ge, 74 Se and 76 Kr isotones, we fixed the value of ζ1,3 = +0.05 for all isotones and vary the ζ2 between (-0.090,-0.084) around the best-fitted to experimental data. It is found that the energy values for the states (

Electric Transition
The E2 transitions calculated using the E2 operator is define as. [7] T (E2) = e π Q π + Q ν e ν (5) The Q π and Q ν operators are defined in equation (2), eπ and eν are boson effective charges for protons and neutrons, depending on the boson number Nρ , these parameters are free and can take any value to fit the experimental data. In present work the effective charge of proton eπ = 0.102(eb) and neutron eν = -0.105 (eb), The reduced electric quadruple probabilities B(E2) is very sensitive to effective charge for proton and neutrons. The B(E2) of 70 Zn, 72 Ge, 74 Se, 76 Kr isotones are shown in the Table:2. It was found that the good agreement between available experimental data and IBM-2 results [7,8,9,10].

Magnetic transition
In order to calculated M1 transition probability must be estimate the effective g-factors for proton and neutron., through used Samba taro relation which can be written as [11,12]: g = (g π N π + g ν N ν )/( N π + N ν ) (6) where g π , g ν are gfactor of nuclear proton and neutron respectively. The total g_factor associated with magnetic momentum, is µ=2g, in this work we used the experimental value of magnetic momentum for the 2 1 + state to estimate the g factor , where µ(2 1 + ) =2g(2 1 + ), it was found that the predicted values are g π =0.552 (µN) and g ν = 0.458 (µN), and (g π -g ν ) = 0.371 (µN). The M1 operator is obtained by making l = 1 in the single boson operator of the IBM-2 and can be written as [13,14].

Mixing Ratios δ (E2/M1)
After calculating the matrix element of B(E2) and B(M1) of gamma transition it is conceivable to look at the strength of E2 and M1 transition in terms of the Multipole mixing ratio (δ) which can be written as follows: [15,16]. δ(E2/M1) 0.835 Eγ Δ(E2/M1) (8) Eγ the transition energy between the two states units (MeV) and Δ(E2/M1) is the ratio between reduced matrix element for E2 and M1 transition which can be written as: [17,18] In  [7,8,9,10]. It is seen that both the magnitude and sign of δ are correctly obtained for most of the selected isotones. Conclusion . In present work we have studied some microscopic properties for 70 Zn, 72 Ge, 74 Se, 76 Kr, isotones and determination some excited energy levels by regulate a small number of parameters in Hamiltonian operators and estimated these parameters, to give the best fit for the energy levels specially for the ground state bands compared with experimental data, while other bands (quasi γ and β band) gave acceptable results because of existence some levels that are undergo mixed symmetry states property (MSS) when rise or drop some energy states compared with corresponding experimental values, due to the mixing wave functions between protons and neutrons, these states are (2 5 + , 3 1 + , 5 1 + , 1 1 + ) gets a quick response when changing majorana term ζ2 as a result of mixture.
Moreover it has been calculated energy ratios 4 1 + 2 1 + ⁄ and 6 1 + 2 1 + ⁄ it was found that these isotones have a varied dynamic symmetries started from rotational motion U(5) least deformed in 70 Zn with increasing deformation of these isotones with increases protons numbers to become transition nuclei between U(5)-O(6) symmetries for 72 Ge, and 74 Se respectively, and finally more closely to O(6) symmetry group for 76 Kr isotones. However have been calculated the probability for electric quadrople transition B(E2), magnetic dipole B(M1) and mixing ratios which All express a good Agreement compared with experimental data.