MIXED SYMMETRY STATES IN 96MO AND

98RU ISOTONES IN THE FRAMEWORK OF

INTERACTING BOSON MODEL

Heiyam Najy Hady*

Kufa University. Education College for girls, Physics department

hiyamn.alkhafaji@uokufa.edu.iq

Ruqaya Talib Kadhim

Kufa University. Education College for girls, Physics department

Reception: 06/12/2022 Acceptance: 28/01/2023 Publication: 15/02/2023

Suggested citation:

N. H., Heiyam and T. K., Ruqaya (2023). Mixed symmetry states in 96Mo and

98Ru isotones in the framework of interacting boson model. 3C Empresa.

Investigación y pensamiento crítico, 12(1), 243-255. https://doi.org/

10.17993/3cemp.2023.120151.243-255

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ABSTRACT

The software package code for interacting boson model-1 and Neutron Proton Boson

code for interacting boson model-2 have been used to calculate energy levels for, for

by estimating a set of parameters which are used to predict the behavior of even-even isotones

within the current scope of work there is clear competition between the two parameters ( and

) in isotones,

as an inverse relationship. This means that vibrational qualities are

continuous mixed with the rotational properties. In interacting boson model-2

parameters( and have been shown similarity with interacting boson model-1

expected. The Majorana parameter effect ( ) on the calculated excitation energy level for

isotones has been accomplished by vary the around the optimum-matches to practical

data. The effect of increasing on mixing symmetry states is the same in all isotones but

different from state to another, we find the state was the lowest mixing

symmetry states still approximately constant in the all. In that time,isotones have

mixed symmetry states, rapidly increasing with increasing . The results of

the calculated energy levels were in acceptable agreement with the experimental data.

There is no pure vibrational property of these isotones

KEYWORDS

Interacting boson model-1&2, MSS's, code and code

PAPER INDEX

IBM

NPBOS

ε,κ,χπ

χν)

ζ2

J+

= 23+

1+, 3+1, 5+1

ζ2

IBM

NPBOS

ABSTRACT

KEYWORDS

INTRODUCTION

INTERACTING BOSON MODEL-1

INTERACTING BOSON MODEL-2

CALCULATIONS AND RESULTS

DISCUSSION AND CONCLUSIONS

REFERENCES

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INTRODUCTION

Even-even isotones considered as medium nuclei mass number, which are always

referred to be as vibrational nuclei, due to the small bosons number outside the

closed N and Z shells, probably what appears from the sequence of energy levels in

the modern experimental decay schemes are stay away from their values of typical

harmonic oscillator (pure vibration), which indicates to energy levels distortion such as

, , and . For this reason, isotones have been re-examined in modern

experimental decay schemes. The interacting boson

model, suggests that the

collective behavior rises from the coupling, through the interaction of the nucleon-

nucleon of the isolated low-lying systems of valence protons and neutrons that is

definite in accordance to the respect of the major shell closure. It is capable of

describing nuclear characteristics such as energies and spins of the levels, decay

probabilities for the emission of gamma quanta, probabilities of electromagnetic

transitions and their reduced matrix elements for different transitions, multipole

moments, and mixing ratios[1-3]. special cases are existed named “dynamic

symmetries”[4-8]. They correspond to the well-known “limits”, vibrational, rotational,

and gamma unsteady nuclei. The concept of dynamic symmetry is of a basic

significance in the IBM, because it permits the exact and analytic solution of the

associated eigenvalue problem for a restricted class of boson Hamiltonians.The

interacting boson model is suitable for describing the low-lying collective states in

even (N,Z) nuclei by a system of interacting s and d-bosons carrying angular

momentum’s and respectively [9].The structure of medium mass nuclei are a

focus of nuclear structure research [10-13]. The structure of nuclei with proton

numbers 42,44 and 46 and the number of neutrons greater than 50 was for many

years a challenge to theoretical explanations because of the fluctuating transition of

nuclear properties between the vibrational features and weak-rotational features

within and [14,16 ].

INTERACTING BOSON MODEL-1

Realizing that there is only one body and two body parts in the interacting boson

paradigm, formation ( ,) and destruction (s,dm) actions are introduced with the

index m=0,±1,±2. When considering on-boson terms in the boson-boson contact, the

greatest Hamiltonian is [1-4].

The boson-boson interacting energy is introduced by are the s and d boson

dynamisms and V. The greatest extensively recycled method of Hamiltonian is

[2-5].

In order to simplify it, sometimes the boson ,vigor value is set to

zero. In addition, the forte of the quadruple points only appears

4+

6+

IBM

0

IBM1

IBM2

s†

d†

H=ε

(s†s)+ε

∑

d†

+V… . (1)

εs,εd

IBM1

H=εnd+a0P†P+a1L.L+a2Q.Q+a3T3T3+a4T4T4…(2)

ε=εd−εs

εs

ε=εd

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is determined by the angular momentum that results from the

interaction of the bosons. Thus, the five and six bosons are lengthy by their solitary

constituent. In cases where the number of bosons is fixed, it is signified by the group

(As for the round oscillator. Correspondingly, it is shaped by the three apertures,

which are , and [14-16], these symmetries are related to the

geometrical idea of the spherical vibrator, deformed rotor and a symmetric ( soft)

deformed rotor, respectively.

INTERACTING BOSON MODEL-2

The interacting boson model-2, is a further step in the development of the

interacting boson model. It is an approach which gives the collective nuclear states as

described by interacting boson -1,a microscopic foundation, since the interacting

boson model-2 can in principle be derived from the shell model. This development,

which is based on the concept of the generalized fermion seniority [1,2], has been

introduced by Arima et. al. [3-8]. The model has given the bosons a direct physical

interpretation as correlated pairs of particles with and .The Hamiltonian

operator in interacting boson model-2 will have three parts: one part for each of proton

and neutron bosons and a third part for describing the proton-neutron

interaction[17-19].

A simple schematic Hamiltonian guided by microscopic consideration is given

by[17-19]:-

where

are proton and neutron energy respectively, they are assumed equal

. The last term in Eq.(4) contains the Majorana operator and it is

usually added in order to remove states of mixed proton neutron symmetry. This term

can be written as[18-21]:-

If there is empirical proof of the existence of the so-called "mixing symmetrical

condition," the Majorana factor is changed in order to adjust the placement of these

levels in the continuum. Due to this approximation, a system of neutron and proton

bosons is taken into consideration. In the interacting boson model2, the microscopic

interpretation of the boson number fixes the total number of bosons, N,

which was previously treated as a parameter in the interacting boson model 1. It is

a0,a1,a2,a3,anda4

U(6)

U(5)

SU(3)

O(6)

γ−

Jπ= 0

Jπ= 2

H=Hπ+ Hν+ Vπν……(3)

H=ε(ndπ+ ndν)+κQπ. Qν+ Vππ + Vνν + Mπν…(4)

Qρ= (d†

ρsρ+s†

ρdρ)2

ρ+χρ(d†

ρdρ)2

ρρ=π,ν…(5)

ρρ =∑

L=0,2,4

2(2L+ 1)1

2Cρ

[

(d†

ρd†

ρ)

(L)

. (dρdρ)(L)

](0)

…(6

)

επ,εν

επ=εν=ε

Mπν

πν =ζ2(s†

νd†

π−d†

νs†

π)

(2)

(

sνdπ−dνsπ

)(2)

+∑

k=1,3

ζk(d†

νd†

π)

(

)

−

(

dνdπ

)(k)

…(7

)

N=Nπ+ Nν

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possible to determine the levels of energy by diagonalizing Hamiltonian Eq. (4) and

experimenting with the parameters and to find the best match to the

observed spectrum. One form of boson can be used to produce spectra that resemble

those of the interacting boson model1 [20,21]. When ,

and

, and( and )are present, respectively. The

U(5) limit, SU(3) limit, and O(6) limits are present. Most nuclei fall halfway between

two of these three limiting instances rather than strictly falling under one of them. The

interactive boson framework enables for a streamlined process between the restrictive

conditions for different isotopes. The systematics of energy ratios of successive levels

of collective bands in even –even medium and heavy mass nuclei were studied for

vibrational and rotational limits for a given band for each I the following ratio were

constructed to define the symmetry of excited band[22].

where denotes the ratio's experimental value. For vibrational

nuclei, the value of energy ratios, r, has vibrates to (0.1r0.35); for transitional nuclei,

(0.4r0.6) and for rotating nuclei, (0.6r1).

CALCULATIONS AND RESULTS

The isotones have neutron number which equivalent(two particles bosons)

and atomic number ( ) respectively, which equivalent ( )

hole proton boson number. By calculating a number of variables specified in the

formulas for the Hamiltonian component equations(2)&(3), the energy levels for have

been calculated using the software packages computer code for

interacting boson model-1 and Neutron Proton Boson for interacting

boson model-2 [5]. Table (1) and Figure (1) provide the anticipated results for the

computations of the stimulated energy state for three isotones that are conducted low-

lying.

ε,κ,χπ,χν

(ε≫κ)

(ε ≪ κ

χπ=χν=−7/2)

ε ≪ κ

χν=− χπ

((J+ 2)

J)=

((J+ 2)

)

exp .−

((J+ 2)

)

vib .

(

(J+ 2)

rot .−

(

(J+ 2)

vib .

((J+ 2)

)

exp .−

((J+ 2)

)

2(J+ 2)/J(J+ 1)

… . (8

)

R((J+ 2)/J)exp .

N= 54

Z= 42and44

4,3and2

IBM −code

NPBOS −code

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Table 1. The parameters have been used in the interacting boson model1&2

Hamiltonian for even-even isotones (in MeV).

IBM1-Parameters in MeV, except χ

98Ru

96Mo

Isotopes

The parameters

0.570.44

0.00.0

0.0080.02

-0.018-0.04

0.0010.001

-0.8-0.8

98Ru

96Mo

Isotopes

The parameters

0.860.9

-0.08-0.08

-0. 8-0. 8

0.010.01

0.0010.001

(-0.4,0.8,0.1)(-0.6,0.8,0.1)

(0.6,0.8,-0.08)(-0.56,0.8,-0.08)

Nπ

-Parameters in MeV, except

IBM 2

ζ1,3

Nν

χν

εd

χν

ζ2

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Figure 1. For even-even isotones, the parameters (in MeV)have been employed in

the interacting boson model-1&2 Hamiltonian as a function of mass numbers

Calculation of energy ratios of ( ), ( ), ( ) and

ratios

for all examined isotones have been calculated and are shown in figure (2).Figure (3)

shows the estimated energy levels for the isotones in comparison to the experimental

data[23-25].

E4+

1/E2+

E6+

1/E2+

E8+

1/E2+

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Figure 2 The experimental [23-25], theoretical and standard [1,2] energy ratios

(, , ) and ratio[22]respectively as a function of mass

numbers for even-even isotones

E4+

1/E2+

E6+

1/E2+

,E8+

1/E2+

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Figure 3 Comparison of the estimated and experimental[23-25] states for isotones,

For isotopes, the effect of Majorana parameters ( , and ) on the levels of the

calculated excitation energy for isotones has been conducted for all by vary the

around the best-fitted with experimental data[23-25] for the states and

. Figure (4) stated the variation of the energy of these states as the Majorana

parameter function .

ζ1,3

ζ2

(2+

3,3+

,5+

1+

ζ2

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Figure 4 Mixed symmetry states in even-even isotones.

DISCUSSION AND CONCLUSIONS

There are two approaches theoretical nuclear models and which are

used to predict the behavior of even-even isotones. The spectra of medium mass

nuclei are usually characterized by the occurrence of low lying collective quadruple

states. In this research, we discuss all results in term of dynamic symmetries, energy

ratio, mixed symmetry states. The relatively medium mass nuclei nearly the mass

number which are located above the double closed and have a

few previous studies. The systematic excitation energy of low lying states mix the

vibrational behavior and rotational behavior as illustrated by energy ratio ( ),

(), ( ) and in figure (2).

In the context of the interacting boson model-1, the competition between the two

parameters ( and ) is observed in the isotones of, where increases in

are

correlated with decreases in

. This indicates that the opposite of the rotational

MSS

IBM1

IBM 2

100

N = 50

Z = 28

E4+

1/E2+

E6+

1/E2+

E8+

1/E2+

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properties, the vibrational features are continuously increasing. They emerge as a

transition between the limits of vibration and rotation. The variants and in

the interacting boson model-2 represent the resemblance with the interacting boson

model-1 anticipated as planned in table (1) and figure (1).

The most significant

element of the interactive boson paradigm is the potential to describe composite

symmetrical conditions in even-even nuclei formed from a combination of the

capabilities of the protons and neutrons waveforms. The lowest MS states are those

with in more vibrational and gamma soft nuclei, while they are detected as

states in rotational nuclei. By varying the Majorana parameter effect )

around the optimum-fitted to experimental data, it has been possible to determine the

calculated excitation energy level for the isotones. It has been discovered that the

state has the lowest mixing symmetry state that is still roughly constant in

all isotones. The effect of raising

on mixing symmetry states is the same in all

isotones but varied from state to state. isotones contain mixed symmetry

states at that moment, which rapidly increase as increases. The collective states

location of mixed proton-neutron symmetry is one of the most remarkable open

experimental difficulties in the study of collective features of nuclei. The experimental

value[23-25] of the first for isotones are represented

spin and parity as ( respectively, somewhat higher than the values in

the best fitting (1.9027and 1.8012 in addition to energy levels values for the

state have a clear mixed symmetry state (

) increasing with increase the

Majorana parameters , the experimental values[23-25] of the first

for isotones

are , very similar to the exact fitted values and

MeV, the same notification applies to

state with an excellent affinity between

practical values and the best fitted.

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ε,κ,χπ

χν

J+= 2+

J+= 1+

(ζ2

J+= 23+

,a2

1+, 3+1, 5+1

ζ2

(2.794and2.406)

MeV

1+and(1+, 2+))

)

MeV

MSS

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