Study of B ( E 2 ) Values of Even-Even Isotopes Using Interacting Boson Model-1

The Electric Reduced Transition Probabilities 2 ↓ have been estimated for the even neutron numbers of isotopes using Interacting Boson Model-1 (IBM-1). The U(5) symmetry, 2 values, intrinsic quadrupole moments and deformation parameters of even neutron = 40 − 46 of isotopes have been studied. The R4/2 values of isotopes have been calculated for the first 4 and 2 energy states and thus U(5) limit is identified.


Introduction
Iachello and Arima developed the Interacting Boson Mode-l (IBM-1) [1,2]. For many-body systems, the vibrational and rotational frequencies characterize the nuclear collective motion of comparable order of magnitude, to prevent a clear-cut distinction between the two types of motion. Moreover, other intermediate situations can occur in nuclei, for example asymmetricrotations and spectra which are neither vibrational nor rotational. A unified description has been proposed for the collective nuclear motion in order to accommodate these facts in terms of a system of interacting bosons [3].
To provide a phenomenological description of spectroscopic data over a wide range of nuclei demonstrating collective features including those customarily interpreted in terms of an-harmonic vibrators or deformed rotors the interacting-boson model (IBM) was found. However, the connection between the parameters of the IBM and the geometrical description has not been clarified, and this has reduced from the method of the IBM. In this research work, an intrinsic state for the IBM has been proposed which may provide the desired connection. States in the IBM are constructed from bosons and bosons which have intrinsic spins 0 and 2 respectively. The corresponding boson creation operators being denoted and respectively, where is the projection quantum number of the boson [4].
The IBM-1 model has been employed theoretically to study the intermediate configuration and configuration mixing around the shell closure Z=28. In IBM-1 and IBMassociated models such as the configuration mixing model in strong connection with the shell model, the conventional collective Hamiltonian approach and the microscopic energydensity function calculated the empirical spectroscopic study within the configuration mixing around the shell. The evolution properties of even-even 104-112 Cd [5], 102-106 Pd [6], 108-112 Pd [7], 100-102 Ru [8] and 120-126 Te [9] have been studied very recently.
The models IBM-1 and IBM-2 are restricted to nuclei with even numbers of protons and neutrons. In order to the fix number of bosons both types of nucleons constitute closed shells with particle numbers 2, 8, 20, 28, 50, 82 and 126 (magic numbers) can be taken. Provided that the protons fill less than half of the furthest shell number of the corresponding active protons has to be divided by two for the boson number attributed to protons. If more than half of the shell is occupied then the boson number reads = (number of holes for protons)/2. By treating the neutrons in an analogous way, one obtains their number of the bosons, ! . In the IBM-1 the boson number N is calculated by adding the partial numbers (number of bosons for the proton and that of the neutron) i.e. = + ! . For example the nucleus # $% && 64 shows the numbers = (54 -50)/2 = 2, ! = (64 -50)/2 = 7 and for # $% & 74 the values = (54 -50)/2 = 2, ! = (82 -74)/2 = 4 hold. Electromagnetic transitions don't alter the boson number but transfers of two identical nucleons lift or lower it by one [10].
In this article, the research work is classified such as: a brief discussion about the IBM-1, quadrupole moments, R 4/2 classification, results and discussions are given in the sections 2, 3, 4 and 5 respectively. Finally the conclusion has been drawn in the section 6.

IBM-1 Method
It is assumed that in the IBM the low-lying collective states are filled with the valance protons and valance neutrons only (i.e. particles outside the major closed shells at 2, 8, 20, 28, 50, 82 and 126), while the closed shell core is inert. Furthermore, it is assumed that the particle configurations are coupled together forming pairs of angular momentum 0 and 2. These proton (neutron) pairs are treated as bosons. Proton (neutron) bosons with angular momentum ' = 0 are denoted by ! and are called s-bosons, while proton (neutron) bosons with angular momentum ' = 2 are denoted by ! and are called d-bosons. The six dimensional unitary groups U(6) of the model gives a simple Hamiltonian, which describes three specific types of collective structure with classical geometrical analogs, these are vibrational U(5), rotational SU (3) and (unstable O(6).In IBM-1 the Hamiltonian of the interacting boson is given by [11,12] ) = ∑ + , + ∑ -,.
Where, + is the intrinsic boson energy and -,. is the interaction between i and j.
Hamiltonian H can be written explicitly in terms of boson creation , and annihilation , ҄ operators such that Where it can be written in the general form as

Quadrupole Moments
The intrinsic quadrupole moments J ? of the nuclei can be written as [9] J ? =^& Here Z is the atomic number, e is the electric charge and i ? = 0.0144j / b is the square of the average radius of the nuclei.

R 4/2 Classification
i %/ is defined by the ratio of the first 4 energy state and the first 2 energy state. It can also be expressed as, There are three types of dynamic symmetries. They are classified into k 5 , mk 3 and o 6 groups. k 5 , mk 3 and o 6 symmetries are called harmonic vibrational, rotational and (-unstable respectively. The symmetries can be identified by using i %/ . If i %/ ≤ 2 then the symmetries will be vibrator symmetries, if the 2 p i %/ p 3 then the symmetries will be ( -unstable symmetries and if i %/ q 3then the symmetries will be rotational symmetries [9].     isotopes. Using equation (7), values of R 4/2 have been calculated for isotopes for even (40-46) neutron numbers. From the figure it is seen that all the R 4/2 values of isotopes are laid bellow the values of 2 as it should be U(5) symmetry [9]. Therefore the isotopes represents the U(5) limit. Figure 2 shows the values of B(E2) ↓ in Weisskopf unit (W.u.) as a function of transition yrast level for isotopes in IBM-1. Using equation (1) the B(E2) ↓ values in W.u. have been calculated and used for isotopes for the transition levels 2 + -0 + , 4 + -2 + , 6 + -4 + and 8 + -6 + .    Figure 4 shows the deformation parameter β as a function of even neutron number of isotopes in IBM-1. The deformation parameter β have been plotted for the isotopes.

Conclusion
The Reduced Transition Probabilities 2 ↓ for the isotopes from 8 → 6 , 6 → 4 , 4 → 2 and 2 → 0 energy states have been estimated by using IBM-1. Using the IBM-1 quadrupole moments and deformation parameters are also calculated. The R 4/2 values have been calculated for the isotopes for the first 4 and 2 energy states and from the values of R 4/2 it is found that these nuclei satisfied the U(5) limit.