Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1)
This paper presents a computational study in the field of nuclear structure by interacting boson model (IBM) to represents very important step formed in the description of collective nuclear excitations and the properties of electromagnetic transition. The ground state energy bands and the reduced t...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2015 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2015
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| Цитувати: | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed // Вопросы атомной науки и техники. — 2015. — № 3. — С. 13-18. — Бібліогр.: 15 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860092709657640960 |
|---|---|
| author | Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. |
| author_facet | Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. |
| citation_txt | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed // Вопросы атомной науки и техники. — 2015. — № 3. — С. 13-18. — Бібліогр.: 15 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | This paper presents a computational study in the field of nuclear structure by interacting boson model (IBM) to represents very important step formed in the description of collective nuclear excitations and the properties of electromagnetic transition. The ground state energy bands and the reduced transition probabilities B(E2) ↓ up to 8₁⁺ level of even-even nuclei ¹¹⁰Pd and ¹¹⁰Cd have been calculated by interacting boson model (IBM-1) and compared with previous experimental values. The set of parameters used in this calculation is the best approximation that has been carried out so far. The ratio of the excitation energies of the first 4⁺ and the first 2⁺ excited states, R₄/₂, is also calculated and an achievable degree of agreement has been investigated in transitional symmetry U(5) - O(6) for ¹¹⁰Cd and O(6) for ¹¹⁰Pd nuclei. We have been compared B(E2) values of ¹¹⁰Pd and ¹¹⁰Cd nuclei with theoretically and experimentally and their systematic studies as a function of angular momentum (L). We have been studied systematically the ratios RL = E(L⁺) / E(2₁⁺) and R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) of those nuclei in the ground-state band. Moreover, we have compared the attention to the analogy between the rotational frequency in ordinary space and Fermi energy in gauge space between ¹¹⁰Pd and ¹¹⁰Cd nuclei.
Представлено компьютерне дослідження в області ядерної структури допомогою моделі взаємодіючих бозонів (IBM-1), яка є дуже важливим кроком в напрямку опису колективних ядерних збуджень і властивості електромагнітних переходів. Рівні енергій основних станів і відповідні вірогідності переходів B(E2) ↓ на рівень 8₁⁺ парно-парного ядра ¹¹⁰Pd і ¹¹⁰Cd були розраховані з допомогою моделі взаємодіючих бозонів (IBM-1) і порівняні з отриманими раніше експериментальними даними. Набір використаних в даній роботі параметрів є найкращим наближенням в порівнянні з отриманими раніше. Відношення енергій збудженняя першого 4⁺ и першого 2⁺ збуджених станів R₄/₂ також вираховані і доступна ступінь узгодження були досліджені в переходній симетріі U(5) - O(6) для ядра ¹¹⁰Cd і O(6) для ядра ¹¹⁰Pd. Ми порівняли B(E2 величини ядер ¹¹⁰Pd і ¹¹⁰Cd з теоретичними і експерименальними і їх систематиними дослідженнями як функцій кутового момента (L) . Ми вивчили систематично відношення RL = E(L⁺) / E(21⁺) і R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) цих ядер в основному стані. Крім того, ми звернули увагу на аналогію між обертовою частотою в звичайному просторі з енергією Фермі в калібровочному просторі між ядрами ¹¹⁰Pd і ¹¹⁰Cd .
Представлено компьютерное исследование в области ядерной структуры с помощью модели взаимодействующих бозонов (IBM-1), представляющей очень важный шаг в направлении описания коллективных ядерных возбуждений и свойств электромагнитных переходов. Уровни энергий основных состояний и соответствующие вероятности переходов B(E2) ↓ на уровень 8₁⁺ четно-четного ядра ¹¹⁰Pd и ¹¹⁰Cd были расчитаны с помощью модели взаимодействующих бозонов (IBM-1) и сравнены с полученными ранее экспериментальными данными. Набор используемых в данной работе параметров является наилучшим приближением в сравнении с полученными ранее. Отношение энергий возбуждения первого 4⁺ и первого 2⁺ возбужденных состояний R₄/₂ также вычислены, и достижимая степень согласия были исследованы в переходной симметрии U(5) - O(6) для ядра ¹¹⁰Cd и O(6) для ядра ¹¹⁰Pd. Мы сравнили B(E2) величины ядер ¹¹⁰Pd и ¹¹⁰Cd с теоретическими и эксперименальными и ихними систематическими исследованиями как функций углового момента (L) . Мы изучили систематически отношения RL = E(L⁺) / E(21⁺) и R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) этих ядер в основном состоянии. Кроме того, мы обратили внимание на аналогию между вращательной частотой в обычном пространстве и энергией Ферми в калибровочном пространстве между ядрами ¹¹⁰Pd и 1¹¹⁰Cd .
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| first_indexed | 2025-12-07T17:24:13Z |
| format | Article |
| fulltext |
NUCLEAR STRUCTURE OF 110Pd AND 110Cd ISOBAR BY
INTERACTING BOSON MODEL (IBM-1)
I.Hossain1∗, HewaY.Abdullah2, I.M.Ahmed3
1Department of Physics, Rabigh College of Science and arts, King Abdulaziz University,
Rabigh 21911, Post box 344, Saudi Arabia;
2Department of Physics, College of Education, Scientific Department, Salahaddin University, Errbil, Krg, Iraq;
3Department of Physics, College of Education, Mosul University, Mosul, Iraq
(Received October 28, 2014)
This paper presents a computational study in the field of nuclear structure by interacting boson model (IBM) to
represents very important step formed in the description of collective nuclear excitations and the properties of
electromagnetic transition. The ground state energy bands and the reduced transition probabilities B(E2) ↓ up to 8+1
level of even-even nuclei 110Pd and 110Cd have been calculated by interacting boson model (IBM-1) and compared
with previous experimental values. The set of parameters used in this calculation is the best approximation that has
been carried out so far. The ratio of the excitation energies of the first 4+ and the first 2+ excited states, R4/2, is
also calculated and an achievable degree of agreement has been investigated in transitional symmetry U(5)−O(6) for
110Cd and O(6) for 110Pd nuclei. We have been compared B(E2) values of 110Pd and 110Cd nuclei with theoretically
and experimentally and their systematic studies as a function of angular momentum (L). We have been studied
systematically the ratios RL = E(L+)/E(2+1 ) and R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+) of those nuclei in
the ground-state band. Moreover, we have compared the attention to the analogy between the rotational frequency
in ordinary space and Fermi energy in gauge space between 110Pd and 110Cd nuclei.
PACS: 21.60.Cs, 23.20Lv, 26.30. +k, 27.60+j
1. INTRODUCTION
Arima and Iachello have developed the interacting
boson model (IBM), which is based on the well-
known shell model and on geometrical collective
model of the atomic nucleus [1,2]. The IBM-1 is
used in the present work to represents very impor-
tant step formed in the description of collective nu-
clear excitations and properties of electromagnetic
transition. The underlying U(6) group structure of
model basis leads to a simple Hamiltonian which is
capable of describing the three specific limits of col-
lective structure vibrational U(5), rotational SU(3)
and gamma unstable O(6). The 110Cd and 110Pd nu-
clei, with two protons and four protons removed from
a strong shell closure, exhibit intriguing aspects of
nuclear structure at low excitation energies, namely
the coexistence and mixing of vibrational or gamma
unstable with other collective degrees of freedom aris-
ing from the promotion of a proton pair across shell
gap [3,4]. The structure of neutron-rich Cd and Pd
isotopes has been studied the subject of many theo-
retical and experimental works in recent years. Long
et al. explained the low-lying levels and high-spin
states of 116, 118, 120Cd in the frame work of inter-
acting boson model [5,6]. The ground state energy
band up to 8+ levels and reduced transition probabil-
ities B(E2) values up to 6+ to 4+ levels in even-even
114−122Cd isotopes were studied under the frame-
work of IBM-1 [7,8]. The evolution properties of
even-even 100−110Pd nuclei were studied by Ahmed
et al. [9]. In this study, we have carried out to com-
pare the nuclear structure of level scheme, reduced
transition probabilities, ground state energy band ra-
tio as function of angular momentum between nuclei
110Pd and 110Cd showing the characteristic U(5) and
O(6) pattern in those low-lying ground state bands
within the frame work of IBM-1.
2. THEORY AND METHOD OF
CALCULATION
2.1. Calculation of Energy levels
The energy levels are calculated using as follows: The
Hamiltonian of the interacting bosons in IBM-1 is
given by Ref.[10].
∗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.13-18.
13
H =
N∑
j=1
εj +
N∑
i<j
Vi,j . (1)
Whereas ε is the intrinsic boson energy and Vij is the
interaction between bosons i and j. The multi-pole
form of the IBM-1 the Hamiltonian is given by Ref.10
H = εnd+a0PP +a1LL+a2QQ+a3T3T3+a4T4T4 .
(2)
The nd operator gives the number of d boson, p is
the pairing operator for the S and d bosons, L is the
angular momentum operator, Q is the quadrupole op-
erator, T3 and T4 are the octupole and hexadecapole
operators, respectively. Moreover a0, a1, a2, and a4
are strength of pairing, angular momentum and mul-
tipole teams. The Hamiltonian as given in Eq.(2)
tends to reduces to three limits, the vibration U(5),
γ-soft O(6) and the rotational SU(3) nuclei [11]. 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 as follows [1,12]:
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)
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 [12]:
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 probabilities using interaction
boson model (IBM-1) [12] 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 B is the boson number, which is equal
half 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 the 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+).
2.3. P -factor
The P -factor is calculated according to Eq.(10).
P =
NnNp
Nn +Np
, (10)
where Nn and Np are the numbers of valence pro-
tons and neutrons, respectively, NnNp represents the
number of p−n interactions and Nn+Np is the num-
ber of pairing interactions.
2.4. Moment of inertia (ϑ) and gamma
energy Eγ
The relation between the moment of inertia (ϑ) and
gamma energy Eγ is given by [9]:
2ϑ
h̄2 =
4I − 2
E(I)− E(I − 2)
=
4I − 2
Eγ
. (11)
And the relation between Eγ and h̄ω is given by
[9,10]:
h̄ω =
E(I)− E(I − 2)√
I(I + 1)−
√
(I − 2)(I − 1)
=
Eγ√
I(I + 1)−
√
(I − 2)(I − 1)
. (12)
2.5. Fermi energy (Gauge space)
The Fermi energies are calculated from the following
relation [13]:
λ(N, I) =
1
2
[Ex(N + 1, I)− Ex(N − 1, I)− SN+1
2n ] ,
(13)
where N is the neutron number between the two even
isotopes which are compared and SN+1
2n is the sepa-
ration energy.
SN+1
2n = EB(Z, N)− EB(Z, N − 2) . (14)
3. RESULTS AND DISCUSSION
3.1. Boson numbers (N)
A simple correlation exists between the nuclei show-
ing identical spectra and their valence neutron pro-
ton (Np), neutron number (Nn). The identical of
such a correlation scheme provided the clue to un-
derstand the identical band phenomena. It was
natural to assume that the nuclei with equal to-
tal boson number Nb = Np + Nn should have the
same deformation and identical spectra. The num-
ber of valance proton Np and neutron Nn has a total
N = (Np + Nn)/2 = nπ + nν bosons The boson
numbers of 110Pd and 110Cd nuclei are 9 and 7 re-
spectively.
14
3.2. P -factor
The pairing interaction between like nucleons drives
the nucleons towards a spherical shape. It forms the
J = 0+ coupling of pairs of identical nucleons that
have spherical symmetric wave functions. Deforma-
tion and collectivity, on the other hand, arise from
configuration mixing which corresponds to a non-
uniform distribution of magnetic sub-state occupa-
tion and hence, of non-spherical shapes. Configura-
tion mixing itself is largely driven by the valence p−n
interaction. Hence it is a pairing p − n competition
that tends to drive the structural evolution of nuclei.
This idea was used to estimate the locus of collectiv-
ity in nuclei. One accepts significant collectivity and
the onset of deformation when the P -factor given ac-
cording to Eq. (10) and values were found 1.71 in
110Cd and 3.11 in 110Pd nuclei.
3.3. The R4/2 classification and ground-state
bands
In the collective dynamics of energies of the even-even
nuclei were grouped into classes, within each class the
ratio:
R4/2 =
E(4+1 )
E(2+1 )
of excitation energies of the first 4+ and the first 2+
excited states. As pointed out by other similar ratios
were characteristics of different collective motions of
the nucleus. R4/2 has a limit value of 2 for vibra-
tional nuclei U(5), 2.5 for γ-unstable nuclei O(6) and
finally 3.33 for rotational nuclei SU(3). The suitable
parameters for each nucleus at the evolving states are
determined using Eq.3 and 4. Table 1 shows the val-
ues of these parameters that have been used to calcu-
late the energy of the yrast-line states for the 110Pd
and 110Cd nuclei. The experimental E(4+1 )/E(2+1 )
for 110Pd and 110Cd Cadmium are 2.46 and 2.34 re-
spectively. Fig.1 shows E(4+1 )/E(2+1 ) values of U(5),
O(6) and SU(3) limit and experimental values of
110Pd and 110Cd nuclei. It is clear that 110Pd and
110Cd nuclei are transitional U(5) − O(6) symme-
try, but 110Pd is very are close to O(6) symmetry.
Table 1. Parameters in (keV ) for even-even 110Pd
and 110Cd nuclei
Nucl. N ϵ K1 K2 K4 K5
110Pd - - - 113.78 -13.55
110Cd 878.79 -46.67 - -22.63 14.79
In Fig.2 we present the energies of the yrast sequences
of ground state band as a function of angular momen-
tum (L) using IBM-1 in 110Pd and 110Cd nuclei and
compared them with previous experimental values
[15]. It is shown that theoretical value using IBM-1
in 110Pd and 110Cd are nicely reproduced to exper-
imental values. The set of parameters used in this
calculation is the best approximation that has been
carried out so far. The excitation levels of ground
state band in 110Cd are greater than those of 110Pd.
Pd Cd
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
E
4
+
/E
2
+
va
lu
e
Nuclei
A=110 SU(3)
O(6)
U(5)
Fig.1. E(4+1 )/E(2+1 ) in experimental values, U(5),
O(6) and SU(3) limit of 110Pd and 110Cd nuclei
-500
0
500
1000
1500
2000
2500
3000
3500
0 2 4 6 8
110Pd (Expt)
110Pd (Th)
110Cd(Expt)
110Cd(Th)
Ex
cit
ati
on
Le
ve
l in
ke
V
Spin
110Cd
110Pd
Fig.2. Ground-state excitation levels as a function
of angular momentum for 110Pd and 110Cd nuclei
3.4. Nuclear collectivity RL = E(L+)/E(2+1 ) of
110Pd and 110Cd
To measure nuclear collectivity, Fig.3 give the com-
parisons of the ratios RL = E(L+)/E(2+1 ) as a func-
tion of angular momentum (L) in the ground-state
band for 110Cd and 110Pd nuclei. The E(L+) indi-
cate ground state energy level at angular momentum
L = 2, 4, 6, and 8. The normalizations were taken
to the energy of their respective 2+1 levels. In Fig.3
it is shown that RL values up to 4+ levels are over-
laps to each other in 110Pd and 110Cd nuclei and
then diverse to the high spin states. We find that
RL values of 110Pd are larger than those of 110Cd
after angular momentum L = 4 and IBM-1 model
show better agreement in 110Pd nucleus than 110Cd
nucleus. The RL values for 110Pd by IBM-1 and
experimental results remain same up to spin 8. How-
ever, we find that the difference RL between the
calculation by IBM-1 and experimental results were
consistently increases after 4+ level and the RL val-
ues were consistently smaller in the IBM calculations
than those in experimental results [14,15] in 110Cd.
15
0
1
2
3
4
5
6
7
0 2 4 6 8 10
110Pd(IBM-1)
110Pd(Expt)
110Cd(IBM-1)
110Cd(Expt)
R L
Spin L
110Pd
110Cd
Fig.3. The yrast sequences of ground state band
of RL = E(L+)/E(2+1 ) as a function of angular
momentum (normalized to the energy of their
respective 2+1 levels) in 110Pd and 110Cd nuclei
Therefore ground-state band by IBM-1 calculation
show better agreement in 110Pd nucleus comparison
to 110Cd nucleus.
3.5. Reduced transition probabilities B(E2)
In the principle, the value of the effective charge α2 of
the IBM-1 was determined by normalizing to the ex-
perimental data B(E2; 2+1 → 0+1 ) of each isotope by
using Eq.(1). From the given experimental value of
the transitions 2 → 0, we have calculated the param-
eter α2
2 for both 110Pd and 110Cd nuclei. The param-
eter α2
2 is useful in order to calculate the transitions
strength (4+ → 2+), (6+ → 4+) and (8+ → 6+). The
B(E2) values were presented in Table 2, where the
previous experimental results [14,15] are compared
with the present calculations.
Table 2. Reduced transition probability B(E2) ↓ in 110Pd and 110Cd isobars
Nuclei Boson ♯ Transition B(E2)Ref [14,15] B(E2)IBM−1
level W.U. e2b2 e2b2
110Pd 9 2+ → 0+ 55.5(9) 0.171(2) 0.171
4+ → 2+ 90(7) 0.277(21) 0.305
6+ → 4+ 108(11) 0.333(34) 0.400
8+ → 6+ 0.457
110Cd 7 2+ → 0+ 27.4(3) 0.084(9) 0.084
4+ → 2+ 0.145
6+ → 4+ 0.182
8+ → 6+ 0.194
0
0.1
0.2
0.3
0.4
0.5
2-0 4-2 6-4 8-6
Pd(Expt)
Pd(Th)
Cd(Ex)
Cd(Th)
B(
E2
) v
al
ue
in
e
2 b2
Ground state gamma transition
110Pd
110Cd
Fig.4. Reduced transition probabilities
B(E2 : 2+ → 0+, 4+ → 2+, 6+ → 4+ and
8+ → 6+) of 110Pd and 110Cd nuclei
The theoretical and experimental results of B(E2)
values were plotted as a function of transition levels
are shown in Fig.4 and it is observed that they are
good agreement within experimental error. The val-
ues of reduced transitional probabilities are greater
in 110Pd than those of 110Cd nucleus. It indicates
that the equivalent effective charge in 110Pd is larger
than that of 110Cd nucleus. The even-even nuclei
110Cd and 110Pd were nicely reproduced by the
experimental results and their fits are satisfactory.
In Fig.5 we Compare the ratio R = B(E2 : L+ →
(L− 2)+)/B(E2 : 2+ → 0+) of IBM-1 and previous
experimental values in the ground state bands (nor-
malized to the B(E2 : 2+ → 0+) as a function of
angular momentum L. It is shown that the results
of R values are increased with increasing of high spin
states. We have found calculated data overlap to ex-
perimental data in 110Cd nucleus. The results of R
values of 110Pd nucleus are consistently smaller by
experimental values than IBM-1 model. However, it
is clear that the calculated results in the present work
are the best agreement with previous results [14,15].
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 110Pd were the main
indicator of gamma soft characters and 110Cd nuclei
indicate the vibration to gamma soft character.
16
0.5
1
1.5
2
2.5
3
2 4 6 8
110Pd(Expt)
110Pd(IBM-1)
110Cd(Expt)
110Cd(IBM-1)
R
Spin
Fig.5. R values of 110Pd and 110Cd nuclei us-
ing IBM-1 and experiment [14,15]. 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 110Pd and 110Cd nuclei
3.6. Moment of inertia of 110Pd and 110Cd
nuclei
The moment of inertia 2ϑ/h̄2 and rotational fre-
quency h̄ω have been calculated from Eq.(11) and
(12) respectively. The ground state bands up to
14 units of angular momentum are investigated for
moment of inertia in 110Pd and 110Cd nuclei. The
moments of inertia as a function of square of rota-
tional energy in 110Pd and 110Cd nuclei are plotted
in Fig.6. In the lowest order according to variable
moment of inertia (VMI) model this should give a
straight line in the plot of inertia 2ϑ/h̄2 as a function
of ω2. It is shown that the value of moment of in-
ertia are greater in 110Pd nucleus than 110Cd in the
lowest order of angular momentum and back-bending
phenomena appear clearly after angular momentum
L = 10 and L = 8 in 110Pd and 110Cd respectively.
0 5 10 15 20 25
0.00
0.02
0.04
0.06
0.08
0.10
0.12
2
2
( ) (keV)
110 cd
110 pd
X 104
Fig.6. Collective moment of inertia vs. square of
rotational energy in 110Pd and 110Cd nuclei
3.7. Fermi energies of 110Pd and 110Cd
The Fermi energies λ(N, I) were calculated from
equation (13). The comparisons of Fermi ener-
gies of 110Pd and 110Cd nuclei in gauge space for
different spin are presented in Fig.7. The Fermi-
energy at 2+, 4+, 6+ and 8+ levels of 110Pd nu-
cleus are −7.500, −7.418, −7.424 and −7.424 keV
respectively. On the other hand the Fermi-energy at
2+, 4+ and 6+ levels of 110Cd are −8.451, −8.410,
and −8.286 keV respectively. It is shown that
Fermi energy as a function of spin for both nu-
clei is similar up to spin 4. The Fermi energy
of 110Pd nuclei remains constant up to spin 8.
0 2 4 6 8 10
-7.25
-7.50
-7.75
-8.00
-8.25
-8.50
-8.75
(N
,I)
K
eV
spin(L)
110Pd
110Cd
Fig.7. Fermi energy vs spin of 110Pd and 110Cd
nuclei
4. CONCLUSIONS
The nuclear structure of ground state band up to pos-
itive parity states 8+ of even-even 110Pd and 110Cd
have been investigated within the frame works of in-
teracting boson model. It was found that the ground
state energy band and electric quadrupole reduced
transition probability by IBM-1 are in good agree-
ment with the previous experimental results [14,15].
The even-even 110Cd and 110Pd nuclei are U(5) −
O(6) and O(6) symmetry respectively. The yrast lev-
els of ground state band are greater in 110Cd than
110Pd nuclei. The reduced transition probabilities
B(E2 : 2+ → 0+, 4+ → 2+, 6+ → 4+ and 8+ → 6+)
for 110Pd are stronger than 110Cd nucleus. Moreover,
the investigation of the back bending phenomena in
ordinary space for even-even 110Pd and 110Cd isobars
were observed and compared with gauge space for the
Fermi energies at different levels.
ACKNOWLEDGEMENTS
The authors are thanks to king Abdulaziz University.
References
1. F. Iachello and A. Arima. The interacting boson
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Phys. 2012, v.E.21(8), p.1250072-(1).
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Hossain, M. K. Kasmin, M. A. Saeed, N. Ibrahim
// Int. J. Modern Phys. 2012, v.E.21(12),
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2012, v.113, p.1315.
ßÄÅÐÍÀß ÑÒÐÓÊÒÓÐÀ ÈÇÎÁÀÐ 110Pd È 110Cd  ÌÎÄÅËÈ
ÂÇÀÈÌÎÄÅÉÑÒÂÓÞÙÈÕ ÁÎÇÎÍÎÂ (IBM − 1)
È.Õîññàéí, Åâàß.Àáäóëëàõ, È.Ì.Àõìåä
Ïðåäñòàâëåíî êîìïüþòåðíîå èññëåäîâàíèå â îáëàñòè ÿäåðíîé ñòðóêòóðû ñ ïîìîùüþ ìîäåëè âçàèìî-
äåéñòâóþùèõ áîçîíîâ (IBM − 1), ïðåäñòàâëÿþùåé î÷åíü âàæíûé øàã â íàïðàâëåíèè îïèñàíèÿ êîë-
ëåêòèâíûõ ÿäåðíûõ âîçáóæäåíèé è ñâîéñòâ ýëåêòðîìàãíèòíûõ ïåðåõîäîâ. Óðîâíè ýíåðãèé îñíîâíûõ
ñîñòîÿíèé è ñîîòâåòñòâóþùèå âåðîÿòíîñòè ïåðåõîäîâ B(E2) ↓ íà óðîâåíü 8+1 ÷åòíî-÷åòíîãî ÿäðà 110Pd
è 110Cd áûëè ðàññ÷èòàíû ñ ïîìîùüþ ìîäåëè âçàèìîäåéñòâóþùèõ áîçîíîâ (IBM − 1) è ñðàâíåíû ñ
ïîëó÷åííûìè ðàíåå ýêñïåðèìåíòàëüíûìè äàííûìè. Íàáîð èñïîëüçóåìûõ â äàííîé ðàáîòå ïàðàìåòðîâ
ÿâëÿåòñÿ íàèëó÷øèì ïðèáëèæåíèåì â ñðàâíåíèè ñ ïîëó÷åííûìè ðàíåå. Îòíîøåíèå ýíåðãèé âîçáóæäå-
íèÿ ïåðâîãî 4+ è ïåðâîãî 2+ âîçáóæäåííûõ ñîñòîÿíèé R4/2 òàêæå âû÷èñëåíû, è äîñòèæèìàÿ ñòåïåíü
ñîãëàñèÿ áûëà èññëåäîâàíà â ïåðåõîäíîé ñèììåòðèè U(5)−O(6) äëÿ ÿäðà 110Cd è O(6) äëÿ ÿäðà 110Pd.
Ìû ñðàâíèëè B(E2) âåëè÷èíû ÿäåð 110Pd è 110Cd ñ òåîðåòè÷åñêèìè è ýêñïåðèìåíòàëüíûìè è èõíèìè
ñèñòåìàòè÷åñêèìè èññëåäîâàíèÿìè êàê ôóíêöèé óãëîâîãî ìîìåíòà (L). Ìû èçó÷èëè ñèñòåìàòè÷åñêè
îòíîøåíèÿ RL = E(L+)/E(2+1 ) è R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+) ýòèõ ÿäåð â îñíîâíîì
ñîñòîÿíèè. Êðîìå òîãî, ìû îáðàòèëè âíèìàíèå íà àíàëîãèþ ìåæäó âðàùàòåëüíîé ÷àñòîòîé â îáû÷íîì
ïðîñòðàíñòâå è ýíåðãèåé Ôåðìè â êàëèáðîâî÷íîì ïðîñòðàíñòâå ìåæäó ÿäðàìè 110Pd è 110Cd.
ßÄÅÐÍÀ ÑÒÐÓÊÒÓÐÀ IÇÎÁÀÐ 110Pd I 110Cd Ó ÌÎÄÅËI ÂÇÀ�ÌÎÄIÞ×ÈÕ
ÁÎÇÎÍIÂ (IBM − 1)
I.Õîññàéí, �âàß.Àáäóëëàõ, I.Ì.Àõìåä
Ïðåäñòàâëåíî êîìï'þòåðíå äîñëiäæåííÿ â îáëàñòi ÿäåðíî¨ ñòðóêòóðè çà äîïîìîãîþ ìîäåëi âçà¹ìîäiþ-
÷èõ áîçîíiâ (IBM−1), ÿêà ¹ äóæå âàæëèâèì êðîêîì ó íàïðÿìêó îïèñó êîëåêòèâíèõ ÿäåðíèõ çáóäæåíü
i âëàñòèâîñòi åëåêòðîìàãíiòíèõ ïåðåõîäiâ. Ðiâíi åíåðãié îñíîâíèõ ñòàíiâ i âiäïîâiäíi âiðîãiäíîñòi ïåðå-
õîäiâ B(E2) ↓ íà ðiâåíü 8+1 ïàðíî-ïàðíîãî ÿäðà 110Pd i 110Cd áóëè ðîçðàõîâàíi çà äîïîìîãîþ ìîäåëi
âçà¹ìîäiþ÷èõ áîçîíiâ (IBM − 1) i ïîðiâíÿíi ç îòðèìàíèìè ðàíiøå åêñïåðèìåíòàëüíèìè äàíèìè. Íàáið
âèêîðèñòàíèõ ó äàíié ðîáîòi ïàðàìåòðiâ ¹ íàéêðàùèì íàáëèæåííÿì ó ïîðiâíÿííi ç îòðèìàíèìè ðàíi-
øå. Âiäíîøåííÿ åíåðãié çáóäæåííÿ ïåðøîãî 4+ i ïåðøîãî 2+ çáóäæåíèõ ñòàíiâ R4/2 òàêîæ âèðàõîâàíi,
i äîñòóïíà ñòóïiíü óçãîäæåííÿ áóëè äîñëiäæåíi â ïåðåõîäíié ñèìåòðii U(5) − O(6) äëÿ ÿäðà 110Cd i
O(6) äëÿ ÿäðà 110Pd. Ìè ïîðiâíÿëè B(E2) âåëè÷èíè ÿäåð 110Pd i 110Cd ç òåîðåòè÷íèìè i åêñïåðè-
ìåíòàëüíèìè i ¨õíiìè ñèñòåìàòè÷íèìè äîñëiäæåííÿìè ÿê ôóíêöié êóòîâîãî ìîìåíòà (L). Ìè âèâ÷èëè
ñèñòåìàòè÷íî âiäíîøåííÿ RL = E(L+)/E(2+1 ) i R = B(E2 : L+ → (L− 2)+)/B(E2 : 2+ → 0+) öèõ ÿäåð
â îñíîâíîìó ñòàíi. Êðiì òîãî, ìè çâåðíóëè óâàãó íà àíàëîãiþ ìiæ îáåðòîâîþ ÷àñòîòîþ â çâè÷àéíîìó
ïðîñòîði ç åíåðãi¹þ Ôåðìi â êàëiáðîâî÷íîìó ïðîñòîði ìiæ ÿäðàìè 110Pd i 110Cd.
18
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| id | nasplib_isofts_kiev_ua-123456789-112110 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:24:13Z |
| publishDate | 2015 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. 2017-01-17T16:41:52Z 2017-01-17T16:41:52Z 2015 Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) / I. Hossain, Hewa Y. Abdullah, I.M. Ahmed // Вопросы атомной науки и техники. — 2015. — № 3. — С. 13-18. — Бібліогр.: 15 назв. — англ. 1562-6016 PACS: 21.60.Cs, 23.20Lv, 26.30. +k, 27.60+j https://nasplib.isofts.kiev.ua/handle/123456789/112110 This paper presents a computational study in the field of nuclear structure by interacting boson model (IBM) to represents very important step formed in the description of collective nuclear excitations and the properties of electromagnetic transition. The ground state energy bands and the reduced transition probabilities B(E2) ↓ up to 8₁⁺ level of even-even nuclei ¹¹⁰Pd and ¹¹⁰Cd have been calculated by interacting boson model (IBM-1) and compared with previous experimental values. The set of parameters used in this calculation is the best approximation that has been carried out so far. The ratio of the excitation energies of the first 4⁺ and the first 2⁺ excited states, R₄/₂, is also calculated and an achievable degree of agreement has been investigated in transitional symmetry U(5) - O(6) for ¹¹⁰Cd and O(6) for ¹¹⁰Pd nuclei. We have been compared B(E2) values of ¹¹⁰Pd and ¹¹⁰Cd nuclei with theoretically and experimentally and their systematic studies as a function of angular momentum (L). We have been studied systematically the ratios RL = E(L⁺) / E(2₁⁺) and R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) of those nuclei in the ground-state band. Moreover, we have compared the attention to the analogy between the rotational frequency in ordinary space and Fermi energy in gauge space between ¹¹⁰Pd and ¹¹⁰Cd nuclei. Представлено компьютерне дослідження в області ядерної структури допомогою моделі взаємодіючих бозонів (IBM-1), яка є дуже важливим кроком в напрямку опису колективних ядерних збуджень і властивості електромагнітних переходів. Рівні енергій основних станів і відповідні вірогідності переходів B(E2) ↓ на рівень 8₁⁺ парно-парного ядра ¹¹⁰Pd і ¹¹⁰Cd були розраховані з допомогою моделі взаємодіючих бозонів (IBM-1) і порівняні з отриманими раніше експериментальними даними. Набір використаних в даній роботі параметрів є найкращим наближенням в порівнянні з отриманими раніше. Відношення енергій збудженняя першого 4⁺ и першого 2⁺ збуджених станів R₄/₂ також вираховані і доступна ступінь узгодження були досліджені в переходній симетріі U(5) - O(6) для ядра ¹¹⁰Cd і O(6) для ядра ¹¹⁰Pd. Ми порівняли B(E2 величини ядер ¹¹⁰Pd і ¹¹⁰Cd з теоретичними і експерименальними і їх систематиними дослідженнями як функцій кутового момента (L) . Ми вивчили систематично відношення RL = E(L⁺) / E(21⁺) і R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) цих ядер в основному стані. Крім того, ми звернули увагу на аналогію між обертовою частотою в звичайному просторі з енергією Фермі в калібровочному просторі між ядрами ¹¹⁰Pd і ¹¹⁰Cd . Представлено компьютерное исследование в области ядерной структуры с помощью модели взаимодействующих бозонов (IBM-1), представляющей очень важный шаг в направлении описания коллективных ядерных возбуждений и свойств электромагнитных переходов. Уровни энергий основных состояний и соответствующие вероятности переходов B(E2) ↓ на уровень 8₁⁺ четно-четного ядра ¹¹⁰Pd и ¹¹⁰Cd были расчитаны с помощью модели взаимодействующих бозонов (IBM-1) и сравнены с полученными ранее экспериментальными данными. Набор используемых в данной работе параметров является наилучшим приближением в сравнении с полученными ранее. Отношение энергий возбуждения первого 4⁺ и первого 2⁺ возбужденных состояний R₄/₂ также вычислены, и достижимая степень согласия были исследованы в переходной симметрии U(5) - O(6) для ядра ¹¹⁰Cd и O(6) для ядра ¹¹⁰Pd. Мы сравнили B(E2) величины ядер ¹¹⁰Pd и ¹¹⁰Cd с теоретическими и эксперименальными и ихними систематическими исследованиями как функций углового момента (L) . Мы изучили систематически отношения RL = E(L⁺) / E(21⁺) и R = B(E2 : L⁺ → (L - 2)⁺) / B(E2 : 2⁺ → 0⁺) этих ядер в основном состоянии. Кроме того, мы обратили внимание на аналогию между вращательной частотой в обычном пространстве и энергией Ферми в калибровочном пространстве между ядрами ¹¹⁰Pd и 1¹¹⁰Cd . The authors are thanks to king Abdulaziz University. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Ядерная физика и элементарные частицы Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) Ядерна структура iзобар ¹¹⁰Pd і ¹¹⁰Cd у моделi взаємодiючих бозонiв (IBM 1) Ядерная структура изобар ¹¹⁰Pd и ¹¹⁰Cd в модели взаимодействующих бозонов (IBM 1) Article published earlier |
| spellingShingle | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) Hossain, I. Abdullah, Hewa Y. Ahmed, I.M. Ядерная физика и элементарные частицы |
| title | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) |
| title_alt | Ядерна структура iзобар ¹¹⁰Pd і ¹¹⁰Cd у моделi взаємодiючих бозонiв (IBM 1) Ядерная структура изобар ¹¹⁰Pd и ¹¹⁰Cd в модели взаимодействующих бозонов (IBM 1) |
| title_full | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) |
| title_fullStr | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) |
| title_full_unstemmed | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) |
| title_short | Nuclear structure of ¹¹⁰Pd and ¹¹⁰Cd isobar by interacting boson model (IBM-1) |
| title_sort | nuclear structure of ¹¹⁰pd and ¹¹⁰cd isobar by interacting boson model (ibm-1) |
| topic | Ядерная физика и элементарные частицы |
| topic_facet | Ядерная физика и элементарные частицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/112110 |
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