Original Article

J Phys Astr, Volume: 5( 2)

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

- *Correspondence:
- 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 Isotopes Using
Interacting Boson Model-1. J Phys Astron. 2017;5(2):114.

### Abstract

The Electric Reduced Transition Probabilities have been estimated for the even neutron numbers of ððð®ð ðð−ðð 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.

### Keywords

*Geometrical; Rotational; Microscopic; Neutron*

### Introduction

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

*d*respectively, where is the projection quantum number of the

^{†}_{μ}*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 ^{104-112}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 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 N_{π} attributed to protons. If more than half of the shell is occupied the boson
number reads N_{π} = ( number of holes for protons )/2. By treating the neutrons in an analogous way, one obtains their number
of bosons N_{v} .In the IBM1 the boson number N is calculated by adding the partial numbers i.e. N=N_{π}+N_{v}.
For example
the nucleus shows the numbers
N_{π}= (54-50)/2=2,N_{v}= (64-50)/2=7 and for the values N_{π}= (54-50)/2=2,N_{v}= (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 s_{π}(s_{v}) and are
called s-bosons, while proton (neutron) bosons with angular momentum *l* = 2are denoted by d_{π}(d_{v}) 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]:

(1)

Where ε is the intrinsic boson energy V_{ij} and is the interaction between i and j.

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

(2)

Where it can be written in the general form as

(3)

Where the total number of *d _{boson}* operator; the pairing operator; the angular momentum operator the octupole and hexadecapole operator; ε=ε

_{d}-ε

_{s}, the boson energy; and the quadrupole operator. The parameters a

_{0}, a

_{1}, a

_{2}, a

_{3}and a

_{4}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 (L

_{i}= 2,4,6,8,……..) are given for anharmonic vibration limit U (5) by [9]

(4)

Where L is the angular momentum and N is the boson number. From the given experimental value of B=(E2)↑ transition the values have been calculated for the transition with the values of Thus the value of parameters α^{2}_{2}(square of effective charge) has been calculated.

**Quadrupole Moments**

The intrinsic quadrupole moments (Q_{0}) of the nuclei can be written as [9]

Here BE2⇑is the upward electromagnetic quadrupole transition probability of the nuclei and e is the electric charge. The upward electromagnetic quadrupole transition probability

and the deformation parameter β can be written as

(6)

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

**R _{4/2} Classification**

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

(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 unstable respectively. We can
identify the symmetries by using R_{4/2}. If R_{4/2} ≤ 2 then the symmetries will be vibrator symmetries, if the then the symmetries will be -unstable symmetries and if R_{4/2}>3 then the symmetries will be rotational symmetries [9].

Neutron number | Energy(KeV) [13, 14] |
|||
---|---|---|---|---|

state | state | |||

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 isotopes.

Nuclide | [15] | Transition level |
Qo(b) | |||
---|---|---|---|---|---|---|

23.46 | 4.69 | 0.25 | 1.58 | |||

8.04 | ||||||

10.05 | ||||||

10.72 | ||||||

10.05 | ||||||

33.1 | 6.62 | 0.254 | 1.85 | |||

11 | ||||||

13.2 | ||||||

13.2 | ||||||

11 | ||||||

6.6 | ||||||

28.61 | 5.72 | 0.258 | 1.70 | |||

9.12 | ||||||

10.26 | ||||||

9.12 | ||||||

5.7 | ||||||

22.4 | 4.48 | 0.263 | 1.47 | |||

6.72 | ||||||

6.72 | ||||||

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 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. 3** shows that the intrinsic quadrupole moment Q_{o} in barn as a function of even neutron number of isotopes
in IBM-1. The intrinsic quadrupole moment Q_{o} in barn have been plotted for the isotopes in IBM-1. The
deformation parameter β have been plotted for the isotopes.

### Conclusion

The Reduced Transition Probabilities B(E2)⇓ for the isotopes from and 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.

### Acknowledgments

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

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