Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1)
In this paper we address the evolution of yrast levels of low-lying structure in the neutron-rich even-even ¹¹⁴⁻¹²²Cd nuclei within the framework of interaction boson model (IBM-1). The reduced transition probabilities B(E2) ↓ between 8₁⁺ to 6₁⁺ states of even-even neutron rich Cd nuclei for N = 66;...
Збережено в:
| Опубліковано в: : | Вопросы атомной науки и техники |
|---|---|
| Дата: | 2015 |
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/112109 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²CD by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed, A.A.Murarak // Вопросы атомной науки и техники. — 2015. — № 3. — С. 19-24. — Бібліогр.: 28 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-112109 |
|---|---|
| record_format |
dspace |
| spelling |
Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. Murarak, A.A. 2017-01-17T16:40:15Z 2017-01-17T16:40:15Z 2015 Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²CD by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed, A.A.Murarak // Вопросы атомной науки и техники. — 2015. — № 3. — С. 19-24. — Бібліогр.: 28 назв. — англ. 1562-6016 PACS: 21.60.Cs, 23.20Lv, 26.30. +k, 27.60+j https://nasplib.isofts.kiev.ua/handle/123456789/112109 In this paper we address the evolution of yrast levels of low-lying structure in the neutron-rich even-even ¹¹⁴⁻¹²²Cd nuclei within the framework of interaction boson model (IBM-1). The reduced transition probabilities B(E2) ↓ between 8₁⁺ to 6₁⁺ states of even-even neutron rich Cd nuclei for N = 66; 68; 70; 72; 74 have been calculated by IBM-1 and compared with the previous available experimental values. The calculated values of ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, and ¹²²Cd, are 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² and 0.149 e²b² respectively. The ratio of the excitation energies of the first 4⁺ and the first 2⁺ excited states, R₄/₂, were also calculated for those nuclei. The ¹¹⁴⁻¹²²Cd isotopes in U(5) - O(6) transitional symmetry were investigated. We have studied the systematic B(E2) values as a function of even neutrons from N = 66 to 74. Furthermore as a measure to quantify the evolution, we have studied systematically the ground state energy ratios RL = E(L⁺) / E(21⁺ ) and transition rate R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) of some of the low-lying quadrupole collective states in comparison to the available experimental data. Представлена еволюція yrast-рівнів низько-лежачої структури в богатому нейтронами парно-парному ядрі ¹¹⁴⁻¹²²Cd у рамках моделі взаємодіючих бозонів (IBM-1). Вірогідності змінених (reduced) переходів B(E2) ↓ між станами 8₁⁺ і 6₁⁺ в богатому нейтронами парно-парному ядрі Cd для N = 66; 68; 70; 72; 74 були розраховані за допомогою IBM-1 і порівняні з раніше відомими експериментальними величинами. Розраховані величини ¹¹⁴Cd, ¹¹⁶Cd, ¹¹⁸Cd, ¹²⁰Cd і ¹²²Cd і дорівнюють 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² та 0.149 e²b², відповідно. Співвідношення енергій збудження першого 4⁺ та першого 2⁺ збуджених станів R₄/₂ були розраховані для цих ядер. ¹¹⁴⁻¹²²Cd ізотопи в U(5) - O(6) симетрії переходів були досліджені. Ми дослідили систематику B(E2) величин, як функцію парних нейтронів від N = 66 до 74. Крім того, вивчаючи кількісно еволюцію, ми систематично вивчили співвідношення енергій основних станів RL = E(L⁺) / E(21⁺ ) та величину R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) для переходів кількох низько-розміщених квадрупольних колективних станів та порівняли з наявними експериментальними даними. Представлена эволюция yrast-уровней низколежащей структуры в богатом нейтронами четно-четном ядре ¹¹⁴⁻¹²²Cd в рамках модели взаимодействующих бозонов (IBM-1). Вероятности измененных (reduced) переходов B(E2) ↓ между состяниями 8₁⁺ и 6₁⁺ в богатом нейтронами четно-четном ядре Cd для N = 66; 68; 70; 72; 74 были расчитаны с помощью IBM-1 и сравнены с ранее известными экспериментальными величинами. Рассчитанные величины ¹¹⁴Cd, ¹¹⁶Cd, ¹¹⁸Cd, ¹²⁰Cd и ¹²²Cd равны 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² и 0.149 e²b², соответственно. Отношения энергий возбуждения первого 4⁺ и первого 2⁺ возбужденных состояний R₄/₂ были рассчитаны для этих ядер. ¹¹⁴⁻¹²²Cd изотопы в U(5) - O(6) симметрии переходов были исследованы. Мы исследовали систематику B(E2) величин, как функцию четных нейтронов от N = 66 до 74. Кроме того, изучая количественно эволюцию, мы систематически изучили отношения энергий основных состояний RL = E(L⁺) / E(21⁺ ) и величину R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) для переходов нескольких низколежащих квадрупольных коллективных состояний и сравнили с доступными экспериментальными данными. Acknowledgement. This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors thankfully acknowledge the technical and financial support of DSR. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) Еволюцiя Yrast -станiв та величини переходiв B(E2 : 8₁⁺ → 6₁⁺ ) ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd у моделi взаємодiючих бозонiв (IBM-1) Эволюция Yrast -состояний и величины переходов B(E2 : 8₁⁺ → 6₁⁺ ) ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd в модели взаимодействующих бозонов (IBM-1) Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) |
| spellingShingle |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. Murarak, A.A. Ядерная физика и элементарные частицы |
| title_short |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) |
| title_full |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) |
| title_fullStr |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) |
| title_full_unstemmed |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd by interacting boson model (IBM-1) |
| title_sort |
evolution of yrast states and b(e2 : 8₁⁺ → 6₁⁺ ) values of ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²cd by interacting boson model (ibm-1) |
| author |
Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. Murarak, A.A. |
| author_facet |
Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. Murarak, A.A. |
| topic |
Ядерная физика и элементарные частицы |
| topic_facet |
Ядерная физика и элементарные частицы |
| publishDate |
2015 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Еволюцiя Yrast -станiв та величини переходiв B(E2 : 8₁⁺ → 6₁⁺ ) ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd у моделi взаємодiючих бозонiв (IBM-1) Эволюция Yrast -состояний и величины переходов B(E2 : 8₁⁺ → 6₁⁺ ) ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²Cd в модели взаимодействующих бозонов (IBM-1) |
| description |
In this paper we address the evolution of yrast levels of low-lying structure in the neutron-rich even-even ¹¹⁴⁻¹²²Cd nuclei within the framework of interaction boson model (IBM-1). The reduced transition probabilities B(E2) ↓ between 8₁⁺ to 6₁⁺ states of even-even neutron rich Cd nuclei for N = 66; 68; 70; 72; 74 have been calculated by IBM-1 and compared with the previous available experimental values. The calculated values of ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, and ¹²²Cd, are 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² and 0.149 e²b² respectively. The ratio of the excitation energies of the first 4⁺ and the first 2⁺ excited states, R₄/₂, were also calculated for those nuclei. The ¹¹⁴⁻¹²²Cd isotopes in U(5) - O(6) transitional symmetry were investigated. We have studied the systematic B(E2) values as a function of even neutrons from N = 66 to 74. Furthermore as a measure to quantify the evolution, we have studied systematically the ground state energy ratios RL = E(L⁺) / E(21⁺ ) and transition rate R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) of some of the low-lying quadrupole collective states in comparison to the available experimental data.
Представлена еволюція yrast-рівнів низько-лежачої структури в богатому нейтронами парно-парному ядрі ¹¹⁴⁻¹²²Cd у рамках моделі взаємодіючих бозонів (IBM-1). Вірогідності змінених (reduced) переходів B(E2) ↓ між станами 8₁⁺ і 6₁⁺ в богатому нейтронами парно-парному ядрі Cd для N = 66; 68; 70; 72; 74 були розраховані за допомогою IBM-1 і порівняні з раніше відомими експериментальними величинами. Розраховані величини ¹¹⁴Cd, ¹¹⁶Cd, ¹¹⁸Cd, ¹²⁰Cd і ¹²²Cd і дорівнюють 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² та 0.149 e²b², відповідно. Співвідношення енергій збудження першого 4⁺ та першого 2⁺ збуджених станів R₄/₂ були розраховані для цих ядер. ¹¹⁴⁻¹²²Cd ізотопи в U(5) - O(6) симетрії переходів були досліджені. Ми дослідили систематику B(E2) величин, як функцію парних нейтронів від N = 66 до 74. Крім того, вивчаючи кількісно еволюцію, ми систематично вивчили співвідношення енергій основних станів RL = E(L⁺) / E(21⁺ ) та величину R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) для переходів кількох низько-розміщених квадрупольних колективних станів та порівняли з наявними експериментальними даними.
Представлена эволюция yrast-уровней низколежащей структуры в богатом нейтронами четно-четном ядре ¹¹⁴⁻¹²²Cd в рамках модели взаимодействующих бозонов (IBM-1). Вероятности измененных (reduced) переходов B(E2) ↓ между состяниями 8₁⁺ и 6₁⁺ в богатом нейтронами четно-четном ядре Cd для N = 66; 68; 70; 72; 74 были расчитаны с помощью IBM-1 и сравнены с ранее известными экспериментальными величинами. Рассчитанные величины ¹¹⁴Cd, ¹¹⁶Cd, ¹¹⁸Cd, ¹²⁰Cd и ¹²²Cd равны 0.272 e²b², 0.281 e²b², 0.259 e²b², 0.190 e²b² и 0.149 e²b², соответственно. Отношения энергий возбуждения первого 4⁺ и первого 2⁺ возбужденных состояний R₄/₂ были рассчитаны для этих ядер. ¹¹⁴⁻¹²²Cd изотопы в U(5) - O(6) симметрии переходов были исследованы. Мы исследовали систематику B(E2) величин, как функцию четных нейтронов от N = 66 до 74. Кроме того, изучая количественно эволюцию, мы систематически изучили отношения энергий основных состояний RL = E(L⁺) / E(21⁺ ) и величину R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) для переходов нескольких низколежащих квадрупольных коллективных состояний и сравнили с доступными экспериментальными данными.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112109 |
| citation_txt |
Evolution of Yrast states and B(E2 : 8₁⁺ → 6₁⁺ ) VALUES OF ¹¹⁴, ¹¹⁶, ¹¹⁸, ¹²⁰, ¹²²CD by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed, A.A.Murarak // Вопросы атомной науки и техники. — 2015. — № 3. — С. 19-24. — Бібліогр.: 28 назв. — англ. |
| work_keys_str_mv |
AT hossaini evolutionofyraststatesandbe28161valuesof114116118120122cdbyinteractingbosonmodelibm1 AT abdullahheway evolutionofyraststatesandbe28161valuesof114116118120122cdbyinteractingbosonmodelibm1 AT ahmedim evolutionofyraststatesandbe28161valuesof114116118120122cdbyinteractingbosonmodelibm1 AT murarakaa evolutionofyraststatesandbe28161valuesof114116118120122cdbyinteractingbosonmodelibm1 AT hossaini evolûciâyraststanivtaveličiniperehodivbe28161114116118120122cdumodelivzaêmodiûčihbozonivibm1 AT abdullahheway evolûciâyraststanivtaveličiniperehodivbe28161114116118120122cdumodelivzaêmodiûčihbozonivibm1 AT ahmedim evolûciâyraststanivtaveličiniperehodivbe28161114116118120122cdumodelivzaêmodiûčihbozonivibm1 AT murarakaa evolûciâyraststanivtaveličiniperehodivbe28161114116118120122cdumodelivzaêmodiûčihbozonivibm1 AT hossaini évolûciâyrastsostoâniiiveličinyperehodovbe28161114116118120122cdvmodelivzaimodeistvuûŝihbozonovibm1 AT abdullahheway évolûciâyrastsostoâniiiveličinyperehodovbe28161114116118120122cdvmodelivzaimodeistvuûŝihbozonovibm1 AT ahmedim évolûciâyrastsostoâniiiveličinyperehodovbe28161114116118120122cdvmodelivzaimodeistvuûŝihbozonovibm1 AT murarakaa évolûciâyrastsostoâniiiveličinyperehodovbe28161114116118120122cdvmodelivzaimodeistvuûŝihbozonovibm1 |
| first_indexed |
2025-11-26T08:10:21Z |
| last_indexed |
2025-11-26T08:10:21Z |
| _version_ |
1850618152298217472 |
| fulltext |
EVOLUTION OF YRAST STATES AND B(E2 : 8+1 → 6+1 )
VALUES OF 114, 116, 118, 120, 122Cd BY INTERACTING BOSON
MODEL (IBM-1)
I.Hossain1∗, HewaY.Abdullah2, I.M.Ahmed3, A.A.Murarak1
1Department of Physics, Rabigh College of Science and arts, King Abdulaziz University;
Rabigh 21911, Post box 344, Saudi Arabia
2Department of Physics, College of Science Education, Salahaddin University, Erbil, Krg, Iraq;
3Department of Physics, College of Education, Mosul University, Mosul, Iraq
(Received March 24, 2014)
In this paper we address the evolution of yrast levels of low-lying structure in the neutron-rich even-even 114−122Cd
nuclei within the framework of interaction boson model (IBM-1). The reduced transition probabilities B(E2) ↓
between 8+1 to 6+1 states of even-even neutron rich Cd nuclei for N = 66, 68, 70, 72, 74 have been calculated by
IBM-1 and compared with the previous available experimental values. The calculated values of 114Cd, 116Cd,
118Cd, 120Cd, and 122Cd, are 0.272 e2b2, 0.281 e2b2, 0.259 e2b2, 0.190 e2b2 and 0.149 e2b2 respectively. The ratio
of the excitation energies of the first 4+ and the first 2+ excited states, R4/2, were also calculated for those nuclei.
The 114−122Cd isotopes in U(5) − O(6) transitional symmetry were investigated. We have studied the system-
atic B(E2) values as a function of even neutrons from N = 66 to 74. Furthermore as a measure to quantify the
evolution, we have studied systematically the ground state energy ratios RL = E(L+)/E(2+1 ) and transition rate
R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+) of some of the low-lying quadrupole collective states in comparison
to the available experimental data.
PACS: 21.60.Cs, 23.20Lv, 26.30. +k, 27.60+j
1. INTRODUCTION
The interacting boson model (IBM-1) is an excellent
interpretive model to understand the nuclear struc-
ture [1,2]. The cadmium nuclei, with two protons
removed from a strong shell closure, exhibit intrigu-
ing aspects of nuclear structure at low excitation
energies, namely the coexistence and mixing of vi-
brational with other collective degrees of freedom
arising from the promotion of a proton pair across
shell gap [3,4]. The structure of neutron-rich Cd iso-
topes has been studied the subject of many theoreti-
cal and experimental works in recent years. The yrast
states up to Iπ = 8+ in N = 48 isotones were found
two-hole states νg−2
9/2 configurations for the N = 50
closed shell. The existences of structure of νg−2
9/2 con-
figurations indicate to find structure of the valance
mirror nuclei πg−2
9/2 configurations [5-8]. Therefore,
it is interesting to study πg−2
9/2 configurations, which
suggest that (πg−2
9/2)
π
I = 0+, 2+, 4+, 6+, 8+ configu-
rations dominate the yrast states and their 8+1 states
are very likely to become isomers.
Moreover B(E2) of the yrast band between 8+1 to
6+1 plays important role in nuclear structure.
There are a number of theoretical works dis-
cussing intruder configuration and configuration mix-
ing in the Cd isotopes. For instance, empirical spec-
troscopic study within the configuration mixing cal-
culation in IBM [9-11], the IBM configuration mixing
model in strong connection with shell model [12,13],
conventional collective Hamiltonian approach [14,15]
and the one starting from self-consistent mean-field
calculation with microscopic energy density func-
tion[16]. Long et al. explained the low-lying lev-
els and high-spin states of 116, 118, 120Cd in the frame
work of interacting boson model [17]. We have cal-
culated the ground state energy band up to 8+1 levels
[18] and reduced transition probabilities B(E2) val-
ues from 6+1 to 4+1 and 4+1 to 2+1 levels in even-even
114−122Cd isotopes by the framework of IBM-1[19].
In this work, we suggest an approach to search for
the dynamical symmetries U(5), SU(3) and O(6) and
to calculate B(E2) values between 8+1 to 6+1 states in
even 114−122Cd isotopes using IBM-1 model [2].
∗Corresponding author: hosain196977@yahoo.com, mihossain@kau.edu.sa Tel.: +966-558141319
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2015, N3(97).
Series: Nuclear Physics Investigations (64), p.19-24.
19
2. THEORY AND METHOD OF
CALCULATION
2.1. Yrast state energy band
The Hamiltonian of the interacting bosons in IBM-1
is given by [2,18].
H =
N∑
j=1
εj +
N∑
i<j
Vi,j , (1)
where ε is the intrinsic boson energy and Vi,j is the
interaction between bosons i and j. In the multi pole
form the Hamiltonian is given by [2,18,19].
H = εnd+a0PP +a1LL+a2QQ+a3T3T3+a4T4T4 .
(2)
Here a0, a1, a2, a3 and a4, are the strength of pair-
ing, the angular momentum and multi pole terms.
The Hamiltonian as given in Eq.(2) tends to reduce
to three limits, the vibration U(5), γ-soft O(6) and
the rotational SU(3) nuclei, starting with the uni-
tary group U(6) and finishing with group O(2). In
U(5) limit, the effective parameter is ε, in the γ-soft
limit, O(6), the effective parameter is the pairing a0,
and in the SU(3) limit, the effective parameter is the
quadrupole a2. The eigenvalues for the three limits
are given by [18].
U(5) : E(nd, ν, L) = εnd +K1nd(nd + 4) +
K4ν(ν + 3) +K5L(L+ 1) . (3)
O(6) : E(σ, τ, L) = K3[N(N + 4)− σ(σ + 4)] +
K4τ(τ + 3) +K5L(L+ 1) . (4)
SU(3) : E(λ, µ, L) = K2[λ
2 + µ2 +
3(λ+ µ) + λµ] +K5L(L+ 1) . (5)
Here, K1, K2, K3, K4 and K5 are other forms of
strength parameters. Many nuclei have a transition
property between two or three of the above limits and
their eigenvalues for the yrast-line are given by [19].
U(5)−O(6) : E(nd, τ, L) = εnd +
K1nd(nd + 4) +K4τ(τ + 3) +K5L(L+ 1) . (6)
U(5)− SU(3) : E(ε, λ, L) = εnd +
K2[λ
2 + 3(λ+ µ)] +K5L(L+ 1) . (7)
O(6)− SU(3) : E(τ, λ, L) = K2[λ
2 +
+3(λ+ µ)] +K4τ(τ + 3) +K5L(L+ 1) . (8)
2.2. Reduced transition probabilities B(E2)
The reduced transition probability in interaction bo-
son model IBM-1 [20] is given by equation (9).
B(E2; J + 2 → J) ↓= α2
2
1
4
(J + 2)(2N − J) . (9)
Where J is the state that the nucleus translates to
it and N is the boson number, which is equal to
half of the number of valence nucleons (proton and
neutrons). The low-lying levels of even-even nuclei
(Ji = 2, 4, 6, 8, ...) usually decay by E2 transition to
the lower-lying yrast level with Jf = Ji − 2. From
the given experimental value of transition (2 → 0),
one can calculate the value the parameter α2
2 for each
isotopes and use this value to calculate the transition
(8+ → 6+).
3. RESULTS AND DISCUSSIONS
The transition from the first excited state to the
ground state is assumed to be a pure E2, (2+ →
0+) transition. The best parameters for ground-
state band in even-even isotopes 114−122Cd are pre-
sented in Table 1. A summary of boson num-
ber, 8+ energy level, gamma-ray transitions 8+ to
6+, experimental B(E2) ↓ between 2+ to ground-
state and reduced transition probabilities between
8+ to 6+ level of even even nuclei from 112Cd to
122Cd, are presented in Table 2. The calculated
results using frame work of IBM-1 are compared
with the previous available experimental results.
Table 1. Parameters in (keV ) for even-even
114−122Cd isotopes [18]
A ε K1 K4 K5
114 768.71 -33.62 -16.40 15.69
116 483.66 26.22 -34.54 14.14
118 484.78 26.29 -30.35 9.17
120 292.83 66.94 -29.95 5.72
122 521.45 55.09 -40.52 1.52
3.1. Boson numbers (N)
A boson represents the pair of valence nucleons and
the boson number is counted as the number of collec-
tive pairs of the valence nucleons. A simple correla-
tion exists between the nuclei showing identical spec-
tra and their valence neutron proton (Np), neutron
number (Nn). The number of valance proton Np and
neutron (Nn) has a total N = (Np+Nn)/2 = nπ+nν
bosons At present 132Sn doubly-magic nucleus is
taken as an inert core to find boson number of 114Cd
to 122Cd nuclei and they are presented in Table 2.
20
Table 2. Reduced transition probability B(E2) ↓ from level 8+1 → 6+1
Nucl. Boson 8+ level⋆ γ Energy ⋆B(E2) B(E2) B(E2)IBM−1
⋆B(E2)
num. in keV (8+ → 6+) (2+ → 0+) (2+ → 0+) (8+ → 6+) (8+ → 6+)
N = nπ + nν keV W.U. e2b2 e2b2 e2b2
114Cd 9=1+8 2669 678 31(19) 0.102 0.272 0.279(71)
116Cd 8=1+7 2824 798 33.5(12) 0.113 0.281
118Cd 7=1+6 2591 771 33(3) 0.113 0.259
120Cd 6=1+5 2886 853 27 0.095 0.190
122Cd 5=1+4 3062 849 26(14) 0.093 0.149
⋆Ref. [6,21-28]
3.2. The R4/2 classification
In the collective dynamics of energies of even-
even nuclei are grouped into classes, within each
class the ratio: R4/2 = E(4+1 )/E(2+1 ) of excita-
tion energies of the first 4+ and the first 2+ ex-
cited states. As pointed out by other similar ra-
tios were characteristics of different collective mo-
tions of the nucleus. An harmonic vibrator has
E(4+1 )/E(2+1 ) = 2.00, an axially symmetric rotor
should have E(4+1 )/E(2+1 ) = 3.33, while X(5) behav-
ior should have E(4+1 )/E(2+1 ) = 2.91. The variation
of the E(4+1 )/E(2+1 ) values as a function of even neu-
tron numbers of Cadmium isotopes for experimental
values, IBM-1, U(5), O(6) and SU(3) limits are
presented in Fig.1. We identified U(5) − O(6) tran-
sitional symmetry in even-even nuclei with Z = 48
and N = 66, 68, 70, 72, 74 with a range 2.04 <
R4/2 < 2.20. But they are near to U(5) symmetry.
1.8
2
2.2
2.4
2.6
2.8
3
3.2
3.4
65 70 75
IBM-1
Expt
U(5)
O(6)
SU(3)
E
4+ /E
2+ v
al
ue
Neutron number
Fig.1. E(4+1 )/E(2+1 ) values as a function of neu-
tron numbers of Cadmium isotopes 114−122Cd for
experimental values, IBM-1, U(5), O(6) and SU(3)
limit
0
1
2
3
4
5
6
65 70 75
IBM-1(E4/E2)
Expt(E4/E2)
IBM-1(E6/E2)
Expt(E6/E2)
IBM-1(E8/E2)
Expt(E8/E2)
E2/E2
R L=
E(
L+ )/E
2+
Neutron number
Fig.2. The yrast sequences of ground state band
of RL = E(L+)/E(2+1 ) as a function of neutron
numbers (normalized to the energy of their respective
2+1 levels) in 114−122Cd nuclei
In Fig.2, we present the energies of the yrast se-
quences of ground state band using IBM-1 (nor-
malized to the energy of their respective 2 levels)
in these nuclei and compared them with previous
experimental values [6, 21-28]. We present the
comparisons of the ratios RL = E(L+)/E(2+1 ) in
the ground-state band (a usually adopted measure
of nuclear collectivity), using the neutron numbers
N = 66, 68, 70, 72, 74. From Figs.1 and 2, we can
see that IBM-1 calculation fit the U(5)−O(6) predic-
tions generally. However, we find that the RL values
are consistently smaller in the IBM calculations than
in experimental values.
3.3. Reduced transition probabilities B(E2)
The values of the reduced transition probabilities
have been fitted the calculated absolute strengths
B(E2) of the transitions within the ground state
band to the experimental ones. The value of the effec-
tive charge α2 of the IBM-1 was determined by nor-
malizing to the experimental data B(E2; 2+1 → 0+1 )
of each isotope by using Eq.(1). From the given
21
experimental value of the transitions (2 → 0), we
calculated value of the parameter α2
2 for each iso-
tope and used this value to calculate the tran-
sition (8+ → 6+). The B(E2) values are pre-
sented in Table 2, where the experimental data is
compared with the present calculations and other
previous work [6, 21-28]. The theoretical and ex-
perimental values of B(E2) values are plotted as
a function of even neutrons represented in Fig.3.
0.1
0.15
0.2
0.25
0.3
0.35
65 70 75
IBM-1
Expt
(8
+
t
o
6+
)
B
(E
2)
e
2
b2
Neutron number
Fig.3. Reduced transition probabilities
B(E2 : 8+ → 6+) as a function of even
neutron numbers of 114−122Cd isotopes
The calculated reduced transition probabilities us-
ing IBM-1 as a function of neutrons number slowly
increase from 0.272 e2b2 to 0.282 e2b2 in the neu-
tron number from 66 to 68, and then decrease to
0.149 e2b2 for up to neutron number 74. In Fig.3,
results of the present work are compared with the
available previous experimental values and shows
good agreement for N = 66. Moreover, the gen-
eral agreement between the calculation and their
previous experimental values for B(E2; 8+ → 6+)
transition in Cd isotopes show a little different for
N = 66. Using IBM-1 in Fig.3, B(E2) values for
the transition 8+ to 6+ decrease smoothly after the
neutron number N = 68. The B(E2) values of the
Z = 48 isotopes with N < 68 differ significantly
from those with N > 68. This difference proba-
bly originates from the orbital occupied by valance
neutron; in the ground state with Z = 48, the va-
lence protons occupy hole-like states in the Z = 50
closed shell, with a main configuration πg−2
9/2. The
valence neutrons occupy mainly particle-like states
in the 50 − 82 shells. Due to the proton-neutron
interaction, the nucleus is deformed. In 114Cd66 and
116Cd68 isotopes the valance neutrons occupy in the
2d3/2 orbitals while for 118,120,122Cd isotopes, the
valance neutrons occupy in the 3S1/21h11/2 orbitals.
The nuclei 114Cd by IBM-1 model nicely repro-
duced the experiment data and were fit satisfactory.
0.5
1
1.5
2
2.5
3
3.5
4
65 70 75
IBM-1(E4/E2)
Expt(E4/E2)
IBM-1(E6/E2)
Expt(E6/E2)
IBM-1(E8/E2)
Expt(E8/E2)
E2/E2
IB
M
-1
(E
4/
E
2)
Neutron number
Fig.4. Comparison of the B(E2) values in
IBM-1 and experimental value. The ratio
R = B(E2 : L+ → (L − 2)+)/B(E2 : 2+ → 0+)
in the ground state bands (normalized to the
B(E2 : 2+ → 0+)) in these nuclei
In Fig.4, we compare the ratio R = B(E2 : L+ →
(L − 2)+)/B(E2 : 2+ → 0+) of IBM-1 and the pre-
vious experimental values in the ground state bands
(normalized to theB(E2 : 2+ → 0+)) as a function of
even neutron number in these nuclei. We found that
the R values were consistently smaller in the IBM
calculations than in the experimental values. How-
ever, we could see that the best agreement is closed
to the calculations with neutron number N = 66.
Actually, in IBM-1 the proton and neutron bosons
are not distinguishable as long as valence protons and
neutrons are both hole-like or both particle-like [2].
The large B(E2) values in 114Cd and 116Cd nuclei
are the main indicator of vibration characters.
4. CONCLUSIONS
The evolution of nuclear low-lying yrast states in the
even neutron-rich 114−122Cd isotopes were investi-
gated within the interaction boson model (IBM-1).
As a measure to quantify the evolution, we calcu-
lated the energy ratios of yrast states and the B(E2)
transition rates of some of the low-lying quadrupole
collective states in comparison to the available exper-
imental data. It is seen that ground state band up to
8+ levels and electric quadrupole reduced transition
probability of those nuclei are in good agreement
with the previous experimental results. The even-
even 114−122Cd isotopes in U(5) − O(6) transitional
symmetry were also investigated.
ACKNOWLEDGEMENTS
Acknowledgement. This work was funded by the
Deanship of Scientific Research (DSR), King Ab-
dulaziz University, Jeddah. Therefore, the authors
thankfully acknowledge the technical and financial
support of DSR.
22
References
1. N.Turkan and I.Maras E(5) // Math. and com-
put. Appli., 2010, v.16, p.428.
2. F. Iachello and A.Arima // The interacting bo-
son model. Cambridge Univ. Press, Cambridge,
1987.
3. R.Kumar, A. Sharma, and J.B.Gupta // Arme-
nian J. Phys. 2010, v.3, p.150.
4. A.Aphahamian, D. S. Brenner, R. F.Casten and
K.Heyda // Phys. Lett. v.B.140, p.22.
5. A.Makishima, M.Asai, T. Ishii, I. Hossain,
M.Ogawa, S. Ichikawa and M. Ishii // Phys. Rev.
1999, v.C.59, p.2331.
6. H.Y.Abdulla, I. Hossain, I.M.Ahmed,
S. T.Ahmed, W.Q.Karwan, M.K.Kasimin,
J. C.Chong, K.K.Viswanathan and N. Ibrahim
// Int J. Phys. Sci. 2011, v.6, p.901.
7. M.Gorska, H.Grawe, D. Foltescu, D. Fossan,
R.Grzywacz, J. Heese, K.Maier M.Rejmund,
H.Roth, R. Schubart // Zeitschrift für Physik A.
Hadrons and Nuclei. 1995, v.353, p.233.
8. N.Mrginean, D.Bucurescu, A.C.Rossi,
L. Skouras, L. Johnstone, D.Bazzacco, S. Lunardi
, de G.Angles, M.Axiotis // Phys. Rev. 2003,
v.C.67(6), p.61310.
9. M. Smbataro // Nucl. Phys. 1982, v.A.380, p.365.
10. M.Deleze et al. // Nucl. Phys. 1993, v.A.551,
p.269.
11. P. E. Garrett, K. L.Green, and J. L.Wood //
Phys. Rev. 2008, v.C.78, p.044307.
12. K.Heyde , P.Van Isacker, Waroquier, and Wenes
// Phys. Rev. 1928, v.C.25, p.3160.
13. P. V. Isacker, S. Pittel, A. Frank, P. D. Duval //
Nucl. Phys. 1986, v.A451, p.202.
14. I. Inci, D.Bonatsos, I. Boztosun // Phys. Rev.
2011, v.C.84, p.024309.
15. D.Bonatsos, P. E.Georgoudis, D. Lenis,
N.Minkov, and C.Quesne // Phys. Rev.
2011, v.C.83, p.044321.
16. L. Prochniak et al. // Int. J. Mod. Phys. 2012,
v.E.21, p.1250036.
17. G. L. Long, S. J. Zhut and H. Z. Sun // J. Phys.
G; Nucl. Part. Phys. 1995, v.21, p.331.
18. I. Hossain, H.Y.Abdullah, I.M.Ahmed,
M.A. Saeed and S.T.Ahmed // Int. J. Modern
Phys. 2012, v.E.21, p.1250072.
19. I. Hossain, H.Y.Abdullah, I.M.Ahmed,
M.A. Saeed and S.T.Ahmed // Armenian
J. Phys. 2012, v.5, p.101.
20. R. F.Casten, D.D.Warner // Rev. Mod. Phys.
1988, v.60, p.389.
21. T.Venkova, W.Andrejtscheff // Atomic Data
and Nucl. Data Tab. 1981, v.26, p.93.
22. MEl-Khoshi // II Nuovo Cimento A. 1993,
v.106, p.875.
23. S.Raman, C.Malakey, W.Milner, C.Nestor //
Atomic data Nucl. Data Tab. 1987, v.36, p.1.
24. J. Blachot // Nucl. data sheets. 2002, v.97, p.593.
25. J. Blachot // Nucl. data sheets. 2010, v.111,
p.717.
26. K.Kitao // Nucl. data sheets. 1995, v.75, p.99.
27. K.Kitao // Nucl. data sheets. 2002, v.96, p.241.
28. Tamuza // Nucl. data sheets. 2007, v.108, p.455.
ÝÂÎËÞÖÈß YRAST-ÑÎÑÒÎßÍÈÉ È ÂÅËÈ×ÈÍÛ ÏÅÐÅÕÎÄΠB(E2 : 8+1 → 6+1 )
114, 116, 118, 120, 122Cd  ÌÎÄÅËÈ ÂÇÀÈÌÎÄÅÉÑÒÂÓÞÙÈÕ ÁÎÇÎÍΠ(IBM-1)
È.Õîññàéí, Åâàß.Àáäóëëàõ, È.Ì.Àõìåä, À.À.Ìóðàðàê
Ïðåäñòàâëåíà ýâîëþöèÿ yrast-óðîâíåé íèçêîëåæàùåé ñòðóêòóðû â áîãàòîì íåéòðîíàìè ÷åòíî-÷åòíîì
ÿäðå 114−122Cd â ðàìêàõ ìîäåëè âçàèìîäåéñòâóþùèõ áîçîíîâ (IBM-1). Âåðîÿòíîñòè èçìåíåííûõ (reduced)
ïåðåõîäîâ B(E2) ↓ ìåæäó ñîñòÿíèÿìè 8+1 è 6+1 â áîãàòîì íåéòðîíàìè ÷åòíî-÷åòíîì ÿäðå Cd äëÿ
N = 66, 68, 70, 72, 74 áûëè ðàñ÷èòàíû ñ ïîìîùüþ IBM-1 è ñðàâíåíû ñ ðàíåå èçâåñòíûìè ýêñïåðèìåí-
òàëüíûìè âåëè÷èíàìè. Ðàññ÷èòàííûå âåëè÷èíû 114Cd, 116Cd, 118Cd, 120Cd è 122Cd ðàâíû 0.272 e2b2,
0.281 e2b2, 0.259 e2b2, 0.190 e2b2 è 0.149 e2b2, ñîîòâåòñòâåííî. Îòíîøåíèå ýíåðãèé âîçáóæäåíèÿ ïåðâîãî
4+ è ïåðâîãî 2+ âîçáóæäåííûõ ñîñòîÿíèé R4/2 áûëè ðàññ÷èòàíû äëÿ ýòèõ ÿäåð. 114−122Cd èçîòîïû
â U(5)−O(6) ñèììåòðèè ïåðåõîäîâ áûëè èññëåäîâàíû. Ìû èññëåäîâàëè ñèñòåìàòèêó B(E2) âåëè÷èí,
êàê ôóíêöèþ ÷åòíûõ íåéòðîíîâ îò N = 66 äî 74. Êðîìå òîãî, èçó÷àÿ êîëè÷åñòâåííî ýâîëþöèþ, ìû
ñèñòåìàòè÷åñêè èçó÷èëè îòíîøåíèÿ ýíåðãèé îñíîâíûõ ñîñòîÿíèé RL = E(L+)/E(2+1 ) è âåëè÷èíó
R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+) äëÿ ïåðåõîäîâ íåñêîëüêèõ íèçêîëåæàùèõ êâàäðóïîëü-
íûõ êîëëåêòèâíûõ ñîñòîÿíèé è ñðàâíèëè ñ äîñòóïíûìè ýêñïåðèìåíòàëüíûìè äàííûìè.
23
ÅÂÎËÞÖIß YRAST-ÑÒÀÍI ÒÀ ÂÅËÈ×ÈÍÈ ÏÅÐÅÕÎÄI B(E2 : 8+1 → 6+1 )
114, 116, 118, 120, 122Cd Ó ÌÎÄÅËI ÂÇÀ�ÌÎÄIÞ×ÈÕ ÁÎÇÎÍI (IBM-1)
I.Õîññàéí, �âàß.Àáäóëëàõ, I.Ì.Àõìåä, À.À.Ìóðàðàê
Ïðåäñòàâëåíà åâîëþöiÿ yrast-ðiâíiâ íèçüêîëåæà÷î¨ ñòðóêòóðè â áàãàòîìó íåéòðîíàìè ïàðíî-ïàðíîìó
ÿäði 114−122Cd ó ðàìêàõ ìîäåëi âçà¹ìîäiþ÷èõ áîçîíiâ (IBM-1). Âiðîãiäíîñòi çìiíåíèõ (reduced) ïå-
ðåõîäiâ B(E2) ↓ ìiæ ñòàíàìè 8+1 i 6+1 ó áàãàòîìó íåéòðîíàìè ïàðíî-ïàðíîìó ÿäði Cd äëÿ N =
66, 68, 70, 72,
74 áóëè ðîçðàõîâàíi çà äîïîìîãîþ IBM-1 i ïîðiâíÿíi ç ðàíiøå âiäîìèìè åêñïåðèìåíòàëüíèìè âåëè-
÷èíàìè. Ðîçðàõîâàíi âåëè÷èíè 114Cd, 116Cd, 118Cd, 120Cd i 122Cd äîðiâíþþòü 0.272 e2b2, 0.281 e2b2,
0.259 e2b2, 0.190 e2b2 òà 0.149 e2b2, âiäïîâiäíî. Ñïiââiäíîøåííÿ åíåðãié çáóäæåííÿ ïåðøîãî 4+ òà ïåð-
øîãî 2+ çáóäæåíèõ ñòàíiâ R4/2 áóëè ðîçðàõîâàíi äëÿ öèõ ÿäåð.
114−122Cd içîòîïè â U(5)−O(6) ñèìåòði¨
ïåðåõîäiâ áóëè äîñëiäæåíi. Ìè äîñëiäèëè ñèñòåìàòèêó B(E2) âåëè÷èí, ÿê ôóíêöiþ ïàðíèõ íåéòðîíiâ
âiä N = 66 äî 74. Êðiì òîãî, âèâ÷àþ÷è êiëüêiñíî åâîëþöiþ, ìè ñèñòåìàòè÷íî âèâ÷èëè ñïiââiäíîøåííÿ
åíåðãié îñíîâíèõ ñòàíiâ RL = E(L+)/E(2+1 ) òà âåëè÷èíó R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+)
äëÿ ïåðåõîäiâ êiëüêîõ íèçüêîðîçìiùåíèõ êâàäðóïîëüíèõ êîëåêòèâíèõ ñòàíiâ òà ïîðiâíÿëè ç íàÿâíèìè
åêñïåðèìåíòàëüíèìè äàíèìè.
24
|