Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation
Interacting Boson Model (IBM-1) is very important for the low-lying combined state in even mass nuclei. We have described the O(6) limit in the ¹⁰⁸’¹¹⁰’¹¹²Ru nuclei by IBM-1 model. The B(E2) strength and energy levels in different bands were established in good agreement theoretically and experiment...
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| Cite this: | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation / I. Hossain, H.H. Kassim, M.A. Al-Jubbori, A. Saleh, K.K. Viswanathan, A. Salam, F.I. Sharrad // Problems of Atomic Science and Technology. — 2023. — № 3. — С. 79-83. — Бібліогр.: 20 назв. — англ. |
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| author | Hossain, I. Kassim, H.H. Al-Jubbori, M.A. Saleh, A. Viswanathan, K.K. Salam, A. Sharrad, F.I. |
| author_facet | Hossain, I. Kassim, H.H. Al-Jubbori, M.A. Saleh, A. Viswanathan, K.K. Salam, A. Sharrad, F.I. |
| citation_txt | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation / I. Hossain, H.H. Kassim, M.A. Al-Jubbori, A. Saleh, K.K. Viswanathan, A. Salam, F.I. Sharrad // Problems of Atomic Science and Technology. — 2023. — № 3. — С. 79-83. — Бібліогр.: 20 назв. — англ. |
| collection | DSpace DC |
| container_title | Problems of Atomic Science and Technology |
| description | Interacting Boson Model (IBM-1) is very important for the low-lying combined state in even mass nuclei. We have described the O(6) limit in the ¹⁰⁸’¹¹⁰’¹¹²Ru nuclei by IBM-1 model. The B(E2) strength and energy levels in different bands were established in good agreement theoretically and experimentally. The calculated potential energy surfaces (PES) of those nuclei were suggested O(6) characters.
Модель взаємодіючих бозонів (IBM-1) дуже важлива для низькорозташованого комбінованого станa в ядрах рівної маси. Ми описали межу O(6) в ядрах ¹⁰⁸’¹¹⁰’¹¹²Ru за моделлю IBM-1. Сила B(E2) і рівні енергії в різних діапазонах були встановлені в хорошому узгодженні теоретично і експериментально. Розраховані поверхні потенціальної енергії (PES) цих ядер були запропоновані O(6) характеру.
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| first_indexed | 2025-12-07T17:02:59Z |
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ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №3(145) 79
https://doi.org/10.46813/2023-145-079
STUDY OF O(6) SYMMETRY IN
108,110,112
Ru ISOTOPES BY IBM-1
CALCULATION
I. Hossain
1*
, Huda H. Kassim
2
, Mushtaq A. Al-Jubbori
3
, Ahmed Saleh
4
, K.K. Viswanathan
5
,
A. Salam
6
, Fadhil I. Sharrad
2,7
1
Department of Physics, Rabigh college of Science & Arts, King Abdulaziz University,
21911 Rabigh, Saudi Arabia;
2
Department of Physics, College of Science, Kerbala University, 56001 Karbala, Iraq;
3
Department of Physics, College of Education for Pure Science, University of Mosul,
41001 Mosul, Iraq;
4
Department of Radiotherapy and Nuclear Medicine, South Egypt Cancer Institute,
Assiut University, Assiut, Egypt;
5
Department of Mathematical Modelling, Faculty of Mathematics, Samarkand State
University, 15, University Boulevard, Samarkand, 140104, Uzbekistan;
6
Department of chemistry, Rabigh college of Science & Arts, King Abdulaziz University,
21911 Rabigh, Saudi Arabia;
7
College of Health and Medical Technology, Alkafeel University, 61001 An Najaf, Iraq
*
E-mail: mihossain@kau.edu.sa
Interacting Boson Model (IBM-1) is very important for the low-lying combined state in even mass nuclei. We
have described the O(6) limit in the
108,110,112
Ru nuclei by IBM-1 model. The B(E2) strength and energy levels in
different bands were established in good agreement theoretically and experimentally. The calculated potential ener-
gy surfaces (PES) of those nuclei were suggested O(6) characters.
PACS: 23.20.-g
INTRODUCTION
Interacting Boson Model (IBM-1) is useful in ex-
plaining the collective nuclear states of even-even nu-
clei. The IBM-1 is established by a fixed number of
bosons (Nb). The neutrons and protons have not been
differentiated in IBM-1. The nuclear structure is com-
plicated because each nucleon interacts with every other
nucleon. Iachello and Arima [1] fruitfully designated the
collective nuclear characters in intermediate mass nuclei
using a system of bosons. Depending on its angular
momentum L, each boson can occupy one of two levels:
s-boson or d-boson. Moreover, this model had generated
the algebra U(6) group: O(6), SU(3) and U(5). Never-
theless, some scientists proposed the construction which
consists of SU(3)–O(6), U(5)–SU(3) and U(5)–
O(6) transitional [4, 5].
The electromagnetic strength, yrast levels and poten-
tial energy surface of Ru isotopes were discussed by
several researchers [6‒11]. Hossain et al. [12, 13] stud-
ied O(6) symmetry of the nuclear structure of
104,106
Ru
by IBM-1 calculation. Sharrad et al. [14] studied the
U(5) symmetry of
96
Ru and
98
Ru nuclei using the inter-
acting boson model (IBM-1). Luo Y. X. et al. [15] stud-
ied Odd-parity bands of
108
Ru,
, 110
Ru and
, 112
Ru iso-
topes. For this it is interesting to study the even-parity
of the ground state band of more neutron-rich nuclei of
Ru (A = 108, 110, 112) by IBM-1 calculation. This
model signifies a vital stage to comprehend the structure
of those nuclei.
The even
108,110,112
Ru isotopes consist of Z = 44 pro-
tons and neutrons N= 64, 66, 68 respectively which fill
the orbits near a double shell closure
100
Sn. The configu-
rations of
108,110,112
Ru isotopes have g
6
2/9
(6 holes of
protons) and g
18,16,14
2/9 (14, 16, and 18 neutrons particle)
near to
100
Sn. It is generally believed that the nuclei far
from a close shell indicate structure from spherical to
deformed symmetry. These neutrons rich deformed nu-
clei will be an excellent topic of interest to be studied
further.
1. METHODOLOGY
The nuclei comprising N nucleons, the IBM model
allocates to occupy a truncated model space. It is re-
sponsible for a numerical clarification of indistinguisha-
ble elements through L = 0 or 2 forming pairs. The
equation of Hamiltonian in IBM-1 is shown as follow
equation:
0 1 2
3 3 3 4 3 4 4
ˆ ˆˆ ˆ ˆ ˆ ˆˆ
ˆ ˆ ˆ ˆ .
dH n a P P a L L a Q Q
a T T a a T T
(1)
The equations of three types of symmetry are given
as follows:
U(5):
1
3 1 4
4 3
( , , ) ( 4)
12
3
( 3)
7 10 70
1
( ) ( 1).
14
d d d d
a
E n L n n n
a a a
a a L L
(2)
80 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №3(145)
SU(3):
2 22
2
1
( , , ) ( 3( ))
2
2
( 1).
8
a
E L
a
a L L
(3)
O(6):
0
3 3
1
( , , ) ( )( 4)
4
( 3) ( 1).
2 10
a
E L N N
a a
a L L
(4)
The Hamiltonian and eigen-values for the three lim-
its are [16],
U(5):
(5) 1 3 3 3
4 4 4
ˆ ˆ ˆ ˆ ˆˆ ( ) ( )
ˆ ˆ( ),
U dH n a L L a T T
a T T
1
4 5
( , , ) ( 4)
( 3) ( 1),
d d d dE n L n K n n
K K L L
(5)
with
1 1 4 1 3 4
5 3 4
1 1 1 3
, ,
12 10 7 70
1 1
.
14 14
K a K a a a
K a a
O(6):
,
3
4 5
( , , ) [ ( 4) ( 4)]
( 3) ( 1),
E L K N N
K K L L
(6)
with
3 0 4 3 5 3 4
1 1 1
, , .
4 2 10
K a K a K a a
SU(3):
(3) 1 2
ˆ ˆˆ ˆ ˆ( ) ( ),UH a L L a Q Q
2 2
2
5
( , , ) ( 3( ))
( 1),
E L K
K L L
(7)
with
2 2 5 1 2
1 3
, .
2 8
K a K a a
2. RESULTS AND DISCUSSION
Using IBM-1 model the results of different types of
energy band, strength of reduced transition probabilities
and potential energy surfaces (PES) of
108, 110, 112
Ru nu-
clei are explained as follows:
2.1. ENERGY BAND
The energy band consists of ground state (G.S.),
gamma and beta band. These are used for the phenome-
nological interaction boson model (IBM-1) to describe
experiential data of
108,110,112
Ru nuclei.
2.1.1. SYMMETRY AND BOSON NUMBER
The symmetry is realistic for those nuclei by
means of available statistics of energy level (E2 : E4: E6 :
E8 = 1.0 : 2.74 : 5.12 : 8.02) [17‒19]. According to dou-
ble shell closure of
100
Sn, the Ru isotopes with N= 64
and 66, have 3 holes of proton boson and 7 and 8 parti-
cles of neutron boson respectively. The
112
Ru nucleus
has 3 hole of proton boson and 7 holes of neutron boson
according to shell closure
132
Si. Therefore total bosons
numbers are 10, 11, and 10 of
108
Ru
and
110
Ru and
112
Ru
respectively. Usually three symmetry U(5), O(6) and
SU(3) depends on
/
≈ 2.00, 2.50 and 3.33
respectively [17‒19]. Here E2
1 and E4
1 are at energy
level 2
1 and 4
1 respectively. The experimental R4/2 of
108
Ru,
110
Ru and
112
Ru isotopes are 2.75, 2.75 and 2.73
respectively which characterized by a non-spherical
spatial distribution of nuclear density and are known as
deformed nuclei gamma soft O(6).
Table 1
Adopted values for the parameters used for IBM-1
calculations. All parameters are given in MeV,
excepted N, and CHQ=0, a2=0, a4=0
a3 a1 a0 N A
0.130 0.010 0.088 10
108
Ru
0.109 0.015 0.094 11
110
Ru
0.092 0.018 0.093 10
112
Ru
The adopted values of parameters were determined
by the equation (4) using the investigational eigen val-
ues E(nd, , L), where nd, and L are quantum num-
bers. The finest fit was occupied up to 12
+
of Ru iso-
topes for the neutron N= 64, 66 and 68. Table 1 show all
parameters (a0, a1, a2a3, a4) are given in MeV.
2.1.2. GROUND STATE BAND
The excitation energy up to 12
+
spins of ground state
band of
108,110,112
Ru [17–19] nuclei are presented in
Fig.1. The 0
+
, 2
+
, 4
+
,6
+
, 8
+
, 10
+
and 12
+
members of the
ground state band of
108
Ru occur at 0, 0.242, 0.665,
1.241, 1.942, 2.741 and 3.529 MeV respectively and
their corresponding value of IBM-1 are 0.0, 0.244,
0.596, 1.057, 1.626, 2.303, and 3.089 MeV. The meas-
ured values of g-band (0
+
,2
+
,4
+
,6
+
,8
+
,10
+
,12
+
) of
110
Ru
and
112
Ru are 0.0, 0.241, 0.664, 1.239, 1.945, 2.759 and
3.647 MeV and 0.0, 0.237, 0.645, 1.189, 1.839, 2.56
and 3.326 MeV respectively. The corresponding theo-
retical calculation of IBM-1 in
110
Ru and
112
Ru are 0.0,
0.241, 0.619, 1.137, 1.792, 2.586 and 3.518 MeV and
0.0 0.236, 0.634, 1.192, 1.911, 2.792 and 3.834 MeV
respectively. It is shown that theoretical and empirical
data establish good agreement and the
108
Ru and
110
Ru
and
112
Ru nuclei are good contenders for O(6) sym-
metry.
0 1 3(6) 3 3. . .OH P P L L T Ta a a
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №3(145) 81
Fig. 1. Ground state (G.S.) band of
108
Ru,
110
Ru and
112
Ru isotopes.IBM-1 and Expt data are indicated by
black and red color respectively
Fig. 2. Gamma band of
108
Ru,
110
Ru and
112
Ru
isotopes. IBM-1 and Expt data are indicated by black
and red color respectively
2.1.3. GAMMA BANDS
The excitation of γ-bands (2
+
, 3
+
, 4
+
….9
+
) is favora-
ble when the potential energy surfaces are found soft for
both beta and gamma deformation parameters. The
bands usually observed at nuclear chart associated with
ellipsoidal nuclear shape. The bands were observed up
to 9
+
(2.844 MeV) in
108
Ru [17], 9
+
(2.777 MeV) in
110
Ru [18] and 8
+
(2.263 MeV) levels in
112
Ru [19]. The
calculated excitation energy in MeV from 2
+
to 9
+
for
gamma bands is compared with previous experimental
data and they are plotted in Fig. 2. From the figures it is
shown that gamma band of all IBM calculations of
those nuclei are in good agreements with available pre-
vious measured data [17–19].
2.1.4. BETA BANDS
The measured members of the beta-vibrational band
observed at 0
+
(0.976 MeV), 2
+
(1.249 MeV),
4
+
(1.644 MeV) and 10
+
(3.149 MeV) in
108
Ru [17] and
0
+
(1.137 MeV) , 2
+
(1.396 MeV), 4
+
( 1.618 MeV) in
110
Ru [18]. There is no measured member of beta band
in
112
Ru. The IBM-1 calculation of bands are present-
ed in Table 2. The excitations energy in MeV for beta
bands (0
+
, 2
+
, 4
+
….10
+
) is compared with available pre-
vious experimental data. It is shown that beta band of all
IBM calculations of those nuclei are in good agreements
with the available previous measured data [17–19].
Table 2
-band (in MeV)
J
π
108
Ru
110
Ru
112
Ru
IBM Exp. IBM Exp. IBM Exp.
0+
0.976 0.976 0.986 1.137 0.831 ‒
2+ 1.219 1.249 1.480 1.396 1.261 ‒
4+ 1.572 1.644 1.605 1.618 1.464 ‒
6+ 2.032 ‒ 2.341 ‒ 2.207 ‒
8+ 2.601 ‒ 3.032 ‒ 2.937 ‒
10+ 3.279 3.150 3.826 ‒ 3.817 ‒
2.2. B (E2) STRENGTHS
The B(E2) strength provides the additional posses-
sions of the construction of nuclei. It is known that the
bosons numbers must be conversed if electric quadruple
transition E2 operator should be a Hermitian tensor of
rank two. The B(E2) strength of
108, 110, 112
Ru in IBM-1
[17–19] is calculated:
2 † † (2) † (2)
2 2
ˆ[ ] [ ]E
BT d s s d d d e Q . (8)
The symbol (s
†
, d
†
) is creation and (s, d) is annihilation
operators for s and d bosons, respectively, although α2
and symbols are two parameters. α2 = effective
charge of boson. and
2
21
( 2, )
2 1
E
i f f i
i
B E J J J T J
J
. (9)
The parameters, α2 and β2 of Eq. (8), accommodated
suitably a set to produce the published B(E2;
).
The calculated effective charge (eB) of those nuclei is
given in Table 3.
Table 3
Effective charge used to reproduced B(E2) values
for even-even
108-112
Ru nuclei
Isotopes N eB (eb)
108
Ru 10 0.082
110
Ru 11 0.079
112
Ru 10 0.092
Table 4
Experimental and the IBM-1 values of B(E2)
for
108,110,112
Ru nuclei in e
2
b
2
J
i -J
f
108Ru 110Ru 112Ru
Exp IBM Exp IBM Exp. IBM
11 02 0.237 0.237 0.207 0.207 0.189 0.189
11 24 ‒ 0.327 0.269 0.286 0.287 0.261
22 24
‒ 0.189 ‒ 0.167 ‒ 0.151
11 46 ‒ 0.362 0.376 0.319 ‒ 0.289
22 46 ‒ 0.250 ‒ 0.224 ‒ 0.199
11 68 ‒ 0.367 ‒ 0.328 ‒ 0.293
11 810 ‒ 0.352 ‒ 0.320 ‒ 0.281
22 810 ‒ 0.259 ‒ 0.243 ‒ 0.207
12 22 ‒ 0.327 ‒ 0.286 ‒ 0.261
12 44 ‒ 0.172 ‒ 0.152 ‒ 0.137
12 66 ‒ 0.117 ‒ 0.104 ‒ 0.093
12 88 ‒ 0.085 ‒ 0.077 ‒ 0.067
The values of eB in unit of eb were estimated to re-
produce experimentally (B(E2;
). The
108,110
Ru
82 ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №3(145)
and
112
Ru isotopes have O(6) character and the values
β2= 0. The calculated and measured strength of B(E2) in
e
2
b
2
for those isotopes are shown in Table 4. We have
normalized the experimental and theoretical data using
B(E2:
) states in each nuclei and then corre-
sponding value up to B(E2: 12 88 ) are given in the
Table 4. The present data of IBM-1 is good with the
obtainable published measured results.
2.3. POTENTIAL ENERGY SURFACE (PES)
The potential energy surface (PES) stretches the evi-
dence for invention microscopic and geometric shapes
of nuclei. The IBM-1 energy surface is created by com-
bining the IBM-1 Hamiltonian’s expectation value with
the coherent state ( ⟩ [20]. The creation opera-
tors (
) act on a state of boson vacuum | ⟩ to produce
the coherent state as follows:
†1
, , ( ) 0N
cN b
N
, (10)
where
† † †
0
2
† †
2 2
1
[cos ( )
1
1
sin ( )] .
2
cb s d
d d
(11)
The EPS can be written in terms of β and γ as
2
4
12 2
3 2
2 3 4
( 1)
( , , ) [
(1 ) (1 )
cos3 ].
dN N N
E N
(12)
The shape of a nucleus could be spherical or distort-
ed depending on whether β=0 or not. The variation in
nucleus symmetry is represented by γ term, when γ = 0,
the nucleus has a prolate shape; when γ=60, it has an
oblate shape. The plot of EPS of even even
108-112
Ru
O(6) Symmetry are shown in Fig. 3.
Fig. 3. Map of potential energy surfaces
(N,, in MeV) of
108,110,112
Ru nuclei
3. CONCLUSIONS
We have used the phenomenological Interaction
Boson Model, IBM-1 to describe the experimental data
for the energies and transition probabilities B(E2) in the
108
Ru,
110
Ru and
112
Ru nuclei. The yrast band, gamma
band, beta band, electromagnetic transition and potential
energy surface of those nuclei have been calculated in
term of O(6) limit of IBM-1. The parameters for excita-
tion up to energy 3.8 MeV of those isotopes were estab-
lished using Hamiltonian of IBM-1. The results of cal-
culation were in acceptable arrangement with the inves-
tigational data. It is established that interacting boson
approximations for those isotopes are suitable to de-
scribe the gamma soft O(6) symmetry.
ACKNOWLEDGEMENTS
Authors are grateful to the authority of DSR for
technical and financial support at King Abdulaziz Uni-
versity, Jeddah. The authors are very grateful to the Col-
lege of Health and Medical Technology, Alkafeel Uni-
ISSN 1562-6016. Problems of Atomic Science and Technology. 2023. №3(145) 83
versity and University of Mosul/College of Education
for Pure Science for their provided facilities, which
helped to improve the quality of this work.
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Article received 04.03.2023
ДОСЛІДЖЕННЯ СИМЕТРІЇ O(6) В ІЗОТОПАХ
108,110,112
Ru ЗА РОЗРАХУНКОМ IBM-1
I. Hossain, Huda H. Kassim, Mushtaq A. Al-Jubbori, Ahmed Saleh, K.K. Вішванатан,
А. Салам, Фадхіл І. Шаррад
Модель взаємодіючих бозонів (IBM-1) дуже важлива для низькорозташованого комбінованого станa в
ядрах рівної маси. Ми описали межу O(6) в ядрах
108,110,112
Ru за моделлю IBM-1. Сила B(E2) і рівні енергії в
різних діапазонах були встановлені в хорошому узгодженні теоретично і експериментально. Розраховані
поверхні потенціальної енергії (PES) цих ядер були запропоновані O(6) характеру.
http://hepnp.ihep.ac.cn/article/app/id/208a510e-33d0-47c5-8c94-d4f606ef48ed/reference
http://hepnp.ihep.ac.cn/article/app/id/208a510e-33d0-47c5-8c94-d4f606ef48ed/reference
https://scholar.google.com/citations?view_op=view_citation&hl=en&user=mnX9kF8AAAAJ&citation_for_view=mnX9kF8AAAAJ:HDshCWvjkbEC
https://scholar.google.com/citations?view_op=view_citation&hl=en&user=mnX9kF8AAAAJ&citation_for_view=mnX9kF8AAAAJ:HDshCWvjkbEC
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| id | nasplib_isofts_kiev_ua-123456789-196145 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:02:59Z |
| publishDate | 2023 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Hossain, I. Kassim, H.H. Al-Jubbori, M.A. Saleh, A. Viswanathan, K.K. Salam, A. Sharrad, F.I. 2023-12-10T16:55:30Z 2023-12-10T16:55:30Z 2023 Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation / I. Hossain, H.H. Kassim, M.A. Al-Jubbori, A. Saleh, K.K. Viswanathan, A. Salam, F.I. Sharrad // Problems of Atomic Science and Technology. — 2023. — № 3. — С. 79-83. — Бібліогр.: 20 назв. — англ. 1562-6016 PACS: 23.20.-g DOI: https://doi.org/10.46813/2023-145-079 https://nasplib.isofts.kiev.ua/handle/123456789/196145 Interacting Boson Model (IBM-1) is very important for the low-lying combined state in even mass nuclei. We have described the O(6) limit in the ¹⁰⁸’¹¹⁰’¹¹²Ru nuclei by IBM-1 model. The B(E2) strength and energy levels in different bands were established in good agreement theoretically and experimentally. The calculated potential energy surfaces (PES) of those nuclei were suggested O(6) characters. Модель взаємодіючих бозонів (IBM-1) дуже важлива для низькорозташованого комбінованого станa в ядрах рівної маси. Ми описали межу O(6) в ядрах ¹⁰⁸’¹¹⁰’¹¹²Ru за моделлю IBM-1. Сила B(E2) і рівні енергії в різних діапазонах були встановлені в хорошому узгодженні теоретично і експериментально. Розраховані поверхні потенціальної енергії (PES) цих ядер були запропоновані O(6) характеру. Authors are grateful to the authority of DSR for technical and financial support at King Abdulaziz University, Jeddah. The authors are very grateful to the College of Health and Medical Technology, Alkafeel University and University of Mosul/College of Education for Pure Science for their provided facilities, which helped to improve the quality of this work. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Problems of Atomic Science and Technology Computational and model systems Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation Дослідження симетрії O(6) в ізотопах ¹⁰⁸’¹¹⁰’¹¹²Ru за розрахунком IBM-1 Article published earlier |
| spellingShingle | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation Hossain, I. Kassim, H.H. Al-Jubbori, M.A. Saleh, A. Viswanathan, K.K. Salam, A. Sharrad, F.I. Computational and model systems |
| title | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation |
| title_alt | Дослідження симетрії O(6) в ізотопах ¹⁰⁸’¹¹⁰’¹¹²Ru за розрахунком IBM-1 |
| title_full | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation |
| title_fullStr | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation |
| title_full_unstemmed | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation |
| title_short | Study of O(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²Ru isotopes by IBM-1 calculation |
| title_sort | study of o(6) symmetry in ¹⁰⁸’¹¹⁰’¹¹²ru isotopes by ibm-1 calculation |
| topic | Computational and model systems |
| topic_facet | Computational and model systems |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/196145 |
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