Original Article
J Phys Astr, Volume: 5( 2)

Study Of B (E2) Values Of Even-Even72-78Ge32 Isotopes Using Interacting Boson Model-1

Islam J, Department of Physics, Mawlana Bhashani Science and Technology University, Satosh,Tangail-1902, Bangladesh, Tel: 0542 236 8558; E-mail: [email protected]

Received: May 26, 2017; Accepted: June 14, 2017; Published: June 20, 2017

Citation: Hasan M, Tabassum M, Roy PK, et al. Study Of B(E2) Values Of Even-Even equation Isotopes Using Interacting Boson Model-1. J Phys Astron. 2017;5(2):114.


The Electric Reduced Transition Probabilities have been estimated for the even neutron numbers of 𝟑𝟐𝑮𝒆 𝟕𝟐−𝟑𝟖 isotopes using Interacting Boson Model-1 (IBM-1). The U(5) symmetry, values, intrinsic quadrupole moments and deformation parameters of even neutron of isotopes have been studied. The R4/2 values of isotopes have been calculated for the first and energy states and thus U(5) limit is identified.


Geometrical; Rotational; Microscopic; Neutron


Iachello and Arima developed the interacting boson model (IBM-1) [1,2]. For many-body systems, thevibrational 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 accomodate 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 anharmonic 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 has been proposed an intrinsic state for the IBM which may provide the desired connection. States in the IBM are constructed from s bosons, with intrinsic spin 0 and d bosons, with intrinsic spin 2, the corresponding boson creation operators being denoted s and dμ respectively, where is the projection quantum number of the d 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 IBM-associated models such as the configuration mixing model in strong connection with the shell model, the conventional collective Hamiltonian approach and the microscopic energy-density function calculated the empirical spectroscopic study within the configuration mixing around the shell. The evolution properties of even-even 104-112Cd [5], 102-106Pd [6], 108-112Pd [7], 100-102Ru [8] and 120-126Te [9] have been studied very recently.

The models IBM1 and IBM2 are restricted to nuclei with even numbers of protons and neutrons. In order to fix the 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 the number of the corresponding active protons has to be divided by two for the boson number Nπ attributed to protons. If more than half of the shell is occupied the boson number reads Nπ = ( number of holes for protons )/2. By treating the neutrons in an analogous way, one obtains their number of bosons Nv .In the IBM1 the boson number N is calculated by adding the partial numbers i.e. N=Nπ+Nv. For example the nucleus shows the numbers Nπ= (54-50)/2=2,Nv= (64-50)/2=7 and for equation the values Nπ= (54-50)/2=2,Nv= (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].

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 L = 0 are denoted by sπ(sv) and are called s-bosons, while proton (neutron) bosons with angular momentum l = 2are denoted by dπ(dv) and are called dbosons. 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 ε is the intrinsic boson energy Vij and is the interaction between i and j.

Hamiltonian H can be written explicitly in terms of boson creation equation and annihilation equation operators such that

equation (2)

Where it can be written in the general form as

equation (3)

Where equation the total number of dboson operator; equationthe pairing operator; equation the angular momentum operator equationthe octupole and hexadecapole operator; ε=εds, the boson energy; and equation the quadrupole operator. The parameters a0, a1, a2, a3and a4 are the strength of the pairing, angular momentum, quadrupole, octupole and hexadecapole interaction respectively between the bosons. The reduced transition probabilities in IBM-1 for even-even nuclei of low-lying levels (Li= 2,4,6,8,……..) are given for anharmonic vibration limit U (5) by [9]

equation (4)

Where L is the angular momentum and N is the boson number. From the given experimental value of B=(E2)↑ transition equation the equation values have been calculated for the transition equation with the values of equation Thus the value of parameters α22(square of effective charge) has been calculated.

Quadrupole Moments

The intrinsic quadrupole moments (Q0) of the nuclei can be written as [9]


Here BE2⇑is the upward electromagnetic quadrupole transition probability of the nuclei and e is the electric charge. The upward electromagnetic quadrupole transition probability equation


and the deformation parameter β can be written as

equation (6)

Here Z is the atomic number, e is the electric charge and equation is the square of the average radius of the nuclei.

R4/2 Classification

The R4/2 is defined by the ratio of the first 4+ energy state and the first 2+ energy state. We can also express it by

equation (7)

There are three types of dynamic symmetries. [TABLEs 1 and 2]. They are classified into U(5) , SU(3) and 0(6) groups. The U(5) ,SU(3) and 0(6) symmetries are called harmonic vibrational, rotational and equation unstable respectively. We can identify the symmetries by using R4/2. If R4/2 ≤ 2 then the symmetries will be vibrator symmetries, if the equation then the symmetries will be -unstable symmetries and if R4/2>3 then the symmetries will be rotational symmetries [9].

  Neutron number Energy(KeV)
[13, 14]
equationstate equationstate
40 1464.094 2464.043 1.68
42 1204.21 2165.26 1.80
44 2503.6 2733.4 1.09
46 1305.3 1498.7 1.15

TABLE 1: The R4/2 values for equation isotopes.

Nuclide equation[15] Transition levelequation
equation equation equation   equationQo(b)
equation 23.46 equation 4.69 0.25 1.58 equation
equation 8.04
equation 10.05
equation 10.72
equation 10.05
equation 33.1 equation 6.62 0.254 1.85 equation
equation 11
equation 13.2
equation 13.2
equation 11
equation 6.6
equation 28.61 equation 5.72 0.258 1.70 equation
equation 9.12
equation 10.26
equation 9.12
equation 5.7
equation 22.4 equation 4.48 0.263 1.47 equation
equation 6.72
equation 6.72
equation 4.48

TABLE 2: The deformation parameter and the quadrupole moment from the IBM-1 of 2+ isotopes.

Results and Discussion

Figure. 1 and Figure. 2 shows the values of B (E2) ↓ in Weisskopf unit (W.u.) as a function of transition yrast level for equation isotopes in IBM-1. Using equation (1) the B (E2) ↓ values in W.u. have been calculated and used for equationisotopes for the transition levels 2+-0+, 4+-2+, 6+-4+ and 8+-6+.


Figure 1: B (E2) ↓ values in W.u against the transition levels for equation isotope.


Figure 2: The intrinsic quadrupole moment Qo in barnvs even neutron number of equation isotopes

Figure. 3 shows that the intrinsic quadrupole moment Qo in barn as a function of even neutron number of equation isotopes in IBM-1. The intrinsic quadrupole moment Qo in barn have been plotted for the equation isotopes in IBM-1. The deformation parameter β have been plotted for the equation isotopes.


Figure 3: The deformation parameter against neutron number.


The Reduced Transition Probabilities B(E2)⇓ for the equation isotopes from equation and equation energy states have been estimated by using IBM-1. Using the IBM-1 quadrupole moments and deformation parameters are also calculated. The R4/2 values have been calculated for the equation isotopes for the first 4+ and 2+ energy states and from the values of R4/2 it is found that these nuclei satisfied the U(5) limit.


Authors thanks to the department of Physics, Mawlana Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh.