Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes
In this paper, the nuclear structure of yrast bands of even-even ¹²⁰⁻¹³⁰Te isotopes has been described as a result of the backbending phenomena of moment of inertia. The square of rotational frequency, moment of inertia and fermi energy of those nuclei has been calculated from available experimental...
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 2014 |
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| Мова: | Англійська |
| Опубліковано: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Цитувати: | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes / M. Hussein, I. Hossain, S.A. Mansour // Вопросы атомной науки и техники. — 2014. — № 5. — С. 27-30. — Бібліогр.: 31 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860183607503486976 |
|---|---|
| author | Hussein, M. Hossainc, I. Mansour, S.A. |
| author_facet | Hussein, M. Hossainc, I. Mansour, S.A. |
| citation_txt | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes / M. Hussein, I. Hossain, S.A. Mansour // Вопросы атомной науки и техники. — 2014. — № 5. — С. 27-30. — Бібліогр.: 31 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | In this paper, the nuclear structure of yrast bands of even-even ¹²⁰⁻¹³⁰Te isotopes has been described as a result of the backbending phenomena of moment of inertia. The square of rotational frequency, moment of inertia and fermi energy of those nuclei has been calculated from available experimental values. We have studied systematic moment of inertia, fermi energy as a function of even neutron number of ¹²⁰⁻¹³⁰Te isotopes. Moment of inertia as a function of the square of the rotational angular velocity for even neutron numbers from N = 68 to 78 in Te isotopes indicates the nature of backbending properties. The investigation of the backbending phenomena in ordinary space for even-even Te isotopes with even neutron number N = 68 to 78 are carried out and compared with gause space. Keywords: Moment of inertia; Te isotopes; Backbending; even-even nuclei; ground state band.
Описана ядерная структура yrast полос (зон) четно-четных изотопов ¹²⁰⁻¹³⁰Te как результат явления backbending момента инерции. Квадрат вращательной частоты, момент инерции и энергия Ферми этих ядер были вычислены из доступных экспериментальных данных. Мы изучили системный момент инерции и энергии Ферми, как функции четного нейтронного числа ¹²⁰⁻¹³⁰Te изотопов. Момент инерции, как функция квадрата угловой скорости вращения для четных чисел нейтронов от N = 68 до 78 в Te изотопах, обнаруживает backbending свойства. Исследования backbending явления в обычном пространстве были выполнены для четно-четных изотопов Te с четными числами нейтронов от N = 68 до 78, и проведены сравнения с результатами для Гауссова пространства.
Описана ядерна структура yrast смуг (зон) парно-парних iзотопiв ¹²⁰⁻¹³⁰Te як наслiдок явища backbending момента iнерцiї. Квадрат обиртової частоти, момент iнерцiї i енергiя Фермi цих ядер були обрахованi iз наявних експериментальних даних. Ми вивчили системний момент iнерцiї i енергiю Фермi, як функцiї парного нейтронного числа ¹²⁰⁻¹³⁰Te iзотопiв. Момент iнерцiї, як функцiя квадрата кутової швидкостi обертання для парних чисел нейтронiв вiд N = 68 до 78 в Te iзотопах, проявляє backbending властивостi. Були вивченi backbending явища для парно-парних iзотопiв Te в звичайному просторi для парних чисел нейтронiв вiд N = 68 до 78, i проведенi порiвняння з даними для Гауссового простору.
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| fulltext |
CHARACTERIZATION OF BACKBENDING IN EVEN-EVEN
120−130Te ISOTOPES
M. Hussein, I. Hossain∗, S.A. Mansour
Department of Physics, Rabigh College of Science and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
(Received November 25, 2013)
In this paper, the nuclear structure of yrast bands of even-even 120−130Te isotopes has been described as a result of
the backbending phenomena of moment of inertia. The square of rotational frequency, moment of inertia and fermi
energy of those nuclei has been calculated from available experimental values. We have studied systematic moment of
inertia, fermi energy as a function of even neutron number of 120−130Te isotopes. Moment of inertia as a function of
the square of the rotational angular velocity for even neutron numbers from N = 68 to 78 in Te isotopes indicates the
nature of backbending properties. The investigation of the backbending phenomena in ordinary space for even-even
Te isotopes with even neutron number N = 68 to 78 are carried out and compared with gause space. Keywords:
Moment of inertia; Te isotopes; Backbending; even-even nuclei; ground state band.
PACS: Nos.: 21.60.-g; 21.10-k
1. INTRODUCTION
The even-even tellurium isotopes Te (Z = 52) have
been extensively investigated both theoretically and
experimentally in recent years with special emphasis
on interpreting experimental data via collective mod-
els. It is known that low-lying collective quadrupole
E2 excitations occur in even-even nuclei Z = 52,
which have been studied both theoretically and ex-
perimentally [1-4]. The electric quadrupole moments
of even 120−128Te isotopes have been studied within
the framework of the semi-microscopic model [5], two-
proton core coupling model [6], dynamic deformation
model [7] and the interacting boson model-2 [8-10].
Discovery of backbending phenomenon at high
spins in the ground state rotational bands of even-
even rare earth nuclei, it has been studied extensively
in many nuclei [1, 12]. The sudden decrease of the
rotational frequency along with anomalous increase
in the moment of inertia have been found to occur in
many deformed nuclei. The word backbending refers
to the phenomenon where a plot of twice the moment
of inertia versus the square of rotational frequency for
various spin states has an S-shaped form [13]. The
variable moment of inertia (VMI) model has a term of
two parameters added to the rotational energy equa-
tion by Mariscottoi for the fitting of energies with
the measured energies [14]. Recently calculation of
the backbending were made using the models, an-
gular momentum projected Tomm-Dancoff approxi-
mation [15], neutron-proton interaction in the frame-
work of the Bardeen-Cooper-Schrieffer (BCS) model
[16], projected shell model [17], and projected con-
figuration interaction (PCI) method in which the de-
formed intrinsic states are directly associated with
shell model wave function [18]. We have earlier stud-
ied evolution properties and backbending properties
of yrast states for even-even 100−110Pd and 104−122Cd
isotopes [19,20]. Electromagnetic reduced transition
probabilities of even-even 104−112Cd [21], 102−106Pd
[22], and 108−112Pd [23], isotopes have been stud-
ied by interacting boson model (IBM − 1). So far
backbending properties of yrast states for even-even
120−130Te isotopes are not found in the literature.
For this purpose, we did an extensive analysis of the
backbending properties of even 120−130Te isotopes by
phenomenological model. The nobility of this work is
to demonstrate the study of extensive analysis of the
backbending properties of even 120−130Te isotopes.
2. THEORETICAL CALCULATION
2.1. Moment of inertia θ and
gamma energy Eγ
The relation between the moment of inertia (ϑ) and
gamma energy Eγ is given by [19]:
2ϑ
h̄2 =
4I − 2
E(I)− E(I − 2)
=
4I − 2
Eγ
. (1)
And the relation between Eγ and h̄ω is given by [24]:
h̄ω =
E(I)− E(I − 2)√
I(I + 1)−
√
(I − 2)(I − 1)
=
Eγ√
I(I + 1)−
√
(I − 2)(I − 1)
. (2)
2.2. Fermi energy (Gauge space)
The Fermi energies are calculated by using the fol-
lowing relation [25]:
λ(N, I) =
1
2
[Ex(N+1, I)−Ex(N−1, I)−SN+1
2n ] , (3)
∗Corresponding author E-mail address: mihossain@kau.edu.sa
26 ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2014, N5 (93).
Series: Nuclear Physics Investigations (63), p.26-30.
where N is the neutron number between the two even
neighboring isotopes which are compared and SN+1
2n
is the separation energy
SN+1
2n = EB(Z,N)− EB(Z,N − 2) . (4)
3. RESULTS AND DISCUSSIONS
Table 1 presents transition level, gamma ray energy,
moment of inertia, square of rotational energy for the
ground state band of even-even 120−130Te isotopes
[26-31].
Table 1. Excitation energies, moment of inertia and square of rotational energy for
even-even 120−126Te isotopes [26-31]
Nucl. I I(I + 1) Eexp(I) , Transition Eγ , 2ϑ/h̄2 , (h̄ω)2 ,
keV Level keV MeV −1 (MeV )2
120Te 2 6 560.4 2+ → 0+ 560.4 10.714 0.0784
4 20 1161.1 4+ → 2+ 601.1 23.294 0.0903
6 42 1775.7 6+ → 4+ 614.6 35.772 0.0946
8 72 2652.4 8+ → 6+ 876.7 34.246 0.1918
10 110 3543.4 10+ → 8+ 891.0 42.648 0.1984
12 156 4459.4 12+ → 10+ 916.0 50.218 0.2097
122Te 2 6 560.4 2+ → 0+ 560.4 10.639 0.0795
4 20 1161.5 4+ → 2+ 601.1 22.690 0.0952
6 42 1776.1 6+ → 4+ 614.6 38.596 0.0812
8 72 2652.8 8+ → 6+ 876.7 41.394 0.2100
10 110 3273.8 10+ → 8+ 621.0 61.191 0.0964
12 156 3978.8 12+ → 10+ 705.0 65.248 0.1243
14 210 4888.8 14+ → 12+ 910 59.34 0.2070
124Te 2 6 602.7 2+ → 0+ 602.7 9.967 0.0906
4 20 1248.6 4+ → 2+ 645.9 21.705 0.1040
6 42 1747.0 6+ → 4+ 498.4 36.756 0.2139
8 72 2664.4 8+ → 6+ 917.4 44.117 0.1156
10 110 3267.1 10+ → 8+ 602.7 86.363 0.0484
126Te 2 6 666.3 2+ → 0+ 666.3 9.009 0.1109
4 20 1361.3 4+ → 2+ 695.0 20.437 0.1173
6 42 1776.2 6+ → 4+ 414.7 53.14 0.0428
8 72 2765.6 8+ → 6+ 989.6 30.333 0.2445
10 110 2973.6 10+ → 8+ 208.0 182.692 0.0108
128Te 2 6 743.2 2+ → 0+ 743.2 8.0753 0.1380
4 20 1497.0 4+ → 2+ 754.8 18.5676 0.1421
6 42 1811.2 6+ → 4+ 314.2 70.0637 0.0246
8 72 2655.4 8+ → 6+ 844.2 35.5450 0.1781
10 110 2756.7 10+ → 8+ 101.3 376.2576 0.0026
12 156 3473.1 12+ → 10+ 717.4 64.1562 0.1285
14 210 4306.8 14+ → 12+ 833.7 64.8259 0.1738
130Te 2 6 839.5 2+ → 0+ 839.5 7.1513 0.1759
4 20 1632.9 4+ → 2+ 793.5 17.6544 0.1572
6 42 1814.3 6+ → 4+ 182.3 120.879 0.0083
8 72 2648.6 8+ → 6+ 833.4 20.367 0.5424
27
Table 2 presents Fermi energy of even-even 120−130Te isotopes [26-31].
Table 2. Fermi energy λ(N, I)MeV of even-even 120−130Te isotopes [26-31]
I 120Te 122Te 124Te 126Te 128Te 130Te
0 -9.420 -9.040 -8.176 -7.847 -7.567 -7.249
2 -9.063 -8.960 -8.157 -7.815 -7.495 -7.201
4 -9.0897 -8.982 -8.162 -7.827 -7.499 -7.229
6 -9.0899 -9.012 -7.999 -8.103 -7.583 -7.315
8 -9.0282 -8.969 -8.293 -7.693 -7.606 -7.598
10 -9.0377 -9.124 -8.267 -7.963
3.1. Moment of inertia
The positive parity yrast levels are connected by a se-
quence of stretched E2 transition with energies which
increase smoothly except around the backbends. The
transition energy ∆EI,I−2 should increase linearly
with I for the constant rotor as ∆EI,I−2 = I/(4I−2)
does not increase, but decrease for certain I values.
The moment of inertia 2ϑ/h̄2 and rotational fre-
quency h̄ω have been calculated from Eq.(1) and (2)
respectively. The ground state bands up to 14 units
of angular momentum are investigated for moment of
inertia in even 120−130Te nuclei. The moments of in-
ertia are plotted versus even neutron number in Fig.1.
Fig.1. Moments of inertia versus even neutron
number N = 68− 78 of 120−130Te
It is shown that 2ϑ/h̄2 as a function of neutrons
do not changes upto spin 4+. 2ϑ/h̄2 as a func-
tion of square of rotational energy in even 120−130Te
nuclei are plotted in Fig.2. In the lowest order ac-
cording to variable moment of inertia (VMI) model
this should give a straight line in the plot of iner-
tia 2ϑ/h̄2 as a function of ω2. It is seen that the
backbending behaviour changes from one isotope
to another in this mass region, the 122−128Te show
backbending at I = 6+ and I = 8+ while 120Te and
130Te do not at all. Moment of inertia are rapidly
increases at 8+ states for N = 72 and 6+ states
for N = 76. Results are presented on collective
∆I = 2 ground band level sequence for the vari-
ation of shapes for Te isotopes with even neutron
from N = 68 − 78. The back-bending phenom-
ena appear clearly in the diagram 2ϑ/h̄2 vs (h̄ω)2.
0,0 0,1 0,2 0,3
0
50
100
150
200
250
120
Te
122
Te
124
Te
126
Te)
128
Te
130
Te)
(MeV)
2
(MeV
-1
)
Fig.2. 2ϑ/h̄2 as a function of square of rotational
energy in even 120−130Te isotopes
3.2. Fermi energies
It is more elucidating to confirm our results in gauge
space plot which show a backbending behaviour when
a change in deformation occurs. The Fermi ener-
gies λ(N, I) of even 120−130Te are calculated from
equation (3). The comparisons of Fermi energies
of Te isotopes with even neutron N = 68 − 78 in
gauge space for different states are presented in Fig.3.
Fig.3. Fermi energies as a function of even neutron
N = 68− 78 of Te isotopes
28
The Fermi-energy of those nuclei at different levels
(2+, 4+, 6+, ...14+) are given in Table 2. The change
of deformation can be explored though the so-called
”gauge-plots” of Fermi energy versus even neutron
number. Fig.3 shows the backbending between the
spherical nuclei with N < 72 and deformed nuclei
with higher neutron number, the λ(N, I) curve re-
flects the level density at the Fermi surface. At
present Gauge-plots show a sudden increase in level
density tends to change the deformation.
4. CONCLUSIONS
Even-even 120−130Te isotopes have drawn the atten-
tion to the analogy between the rotational frequency
in ordinary space and Fermi energy in gauge space.
The investigation of the backbending phenomena in
ordinary space for even-even 120−130Te isotopes are
observed and compared with gauge space for the
Fermi energies up to levels 14+. Results are ex-
tremely useful for compiling nuclear data table.
ACKNOWLEDGEMENTS
This work was funded by the Deanship of Scientific
Research (DSR), King Abdulaziz University, Jeddah,
under Grant No.(662-016-D1434). Therefore, the au-
thors thankfully acknowledge the technical and finan-
cial support of DSR.
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29
ÕÀÐÀÊÒÅÐÈÇÀÖÈß ßÂËÅÍÈß BACKBENDING  ×ÅÒÍÎ-×ÅÒÍÛÕ ÈÇÎÒÎÏÀÕ
120−130Te
Ì. Õóññåéí, È. Õîññàéí, Ñ.À. Ìàíñîóð
Îïèñàíà ÿäåðíàÿ ñòðóêòóðà yrast ïîëîñ (çîí) ÷åòíî-÷åòíûõ èçîòîïîâ 120−130Te êàê ðåçóëüòàò ÿâëå-
íèÿ backbending ìîìåíòà èíåðöèè. Êâàäðàò âðàùàòåëüíîé ÷àñòîòû, ìîìåíò èíåðöèè è ýíåðãèÿ Ôåðìè
ýòèõ ÿäåð áûëè âû÷èñëåíû èç äîñòóïíûõ ýêñïåðèìåíòàëüíûõ äàííûõ. Ìû èçó÷èëè ñèñòåìíûé ìîìåíò
èíåðöèè è ýíåðãèè Ôåðìè, êàê ôóíêöèè ÷åòíîãî íåéòðîííîãî ÷èñëà 120−130Te èçîòîïîâ. Ìîìåíò èíåð-
öèè, êàê ôóíêöèÿ êâàäðàòà óãëîâîé ñêîðîñòè âðàùåíèÿ äëÿ ÷åòíûõ ÷èñåë íåéòðîíîâ îò N = 68 äî
78 â Te èçîòîïàõ, îáíàðóæèâàåò backbending ñâîéñòâà. Èññëåäîâàíèÿ backbending ÿâëåíèÿ â îáû÷íîì
ïðîñòðàíñòâå áûëè âûïîëíåíû äëÿ ÷åòíî-÷åòíûõ èçîòîïîâ Te ñ ÷åòíûìè ÷èñëàìè íåéòðîíîâ îò N = 68
äî 78, è ïðîâåäåíû ñðàâíåíèÿ ñ ðåçóëüòàòàìè äëÿ Ãàóññîâà ïðîñòðàíñòâà.
ÕÀÐÀÊÒÅÐÈÇÀÖIß ßÂÈÙÀ BACKBENDING  ÏÀÐÍÎ-ÏÀÐÍÈÕ IÇÎÒÎÏÀÕ
120−130Te
Ì. Õóññåéí, I. Õîññàéí, Ñ.À. Ìàíñîóð
Îïèñàíà ÿäåðíà ñòðóêòóðà yrast ñìóã (çîí) ïàðíî-ïàðíèõ içîòîïiâ 120−130Te ÿê íàñëiäîê ÿâèùà backbending
ìîìåíòà iíåðöi¨. Êâàäðàò îáèðòîâî¨ ÷àñòîòè, ìîìåíò iíåðöi¨ i åíåðãiÿ Ôåðìi öèõ ÿäåð áóëè îáðàõîâàíi iç
íàÿâíèõ åêñïåðèìåíòàëüíèõ äàíèõ. Ìè âèâ÷èëè ñèñòåìíèé ìîìåíò iíåðöi¨ i åíåðãiþ Ôåðìi, ÿê ôóíêöi¨
ïàðíîãî íåéòðîííîãî ÷èñëà 120−130Te içîòîïiâ. Ìîìåíò iíåðöi¨, ÿê ôóíêöiÿ êâàäðàòà êóòîâî¨ øâèäêîñòi
îáåðòàííÿ äëÿ ïàðíèõ ÷èñåë íåéòðîíiâ âiä N = 68 äî 78 â Te içîòîïàõ, ïðîÿâëÿ¹ backbending âëàñòèâî-
ñòi. Áóëè âèâ÷åíi backbending ÿâèùà äëÿ ïàðíî-ïàðíèõ içîòîïiâ Te â çâè÷àéíîìó ïðîñòîði äëÿ ïàðíèõ
÷èñåë íåéòðîíiâ âiä N = 68 äî 78, i ïðîâåäåíi ïîðiâíÿííÿ ç äàíèìè äëÿ Ãàóññîâîãî ïðîñòîðó.
30
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| id | nasplib_isofts_kiev_ua-123456789-80499 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T18:03:38Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Hussein, M. Hossainc, I. Mansour, S.A. 2015-04-18T15:24:01Z 2015-04-18T15:24:01Z 2014 Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes / M. Hussein, I. Hossain, S.A. Mansour // Вопросы атомной науки и техники. — 2014. — № 5. — С. 27-30. — Бібліогр.: 31 назв. — англ. 1562-6016 PACS: Nos.: 21.60.-g; 21.10-k https://nasplib.isofts.kiev.ua/handle/123456789/80499 In this paper, the nuclear structure of yrast bands of even-even ¹²⁰⁻¹³⁰Te isotopes has been described as a result of the backbending phenomena of moment of inertia. The square of rotational frequency, moment of inertia and fermi energy of those nuclei has been calculated from available experimental values. We have studied systematic moment of inertia, fermi energy as a function of even neutron number of ¹²⁰⁻¹³⁰Te isotopes. Moment of inertia as a function of the square of the rotational angular velocity for even neutron numbers from N = 68 to 78 in Te isotopes indicates the nature of backbending properties. The investigation of the backbending phenomena in ordinary space for even-even Te isotopes with even neutron number N = 68 to 78 are carried out and compared with gause space. Keywords: Moment of inertia; Te isotopes; Backbending; even-even nuclei; ground state band. Описана ядерная структура yrast полос (зон) четно-четных изотопов ¹²⁰⁻¹³⁰Te как результат явления backbending момента инерции. Квадрат вращательной частоты, момент инерции и энергия Ферми этих ядер были вычислены из доступных экспериментальных данных. Мы изучили системный момент инерции и энергии Ферми, как функции четного нейтронного числа ¹²⁰⁻¹³⁰Te изотопов. Момент инерции, как функция квадрата угловой скорости вращения для четных чисел нейтронов от N = 68 до 78 в Te изотопах, обнаруживает backbending свойства. Исследования backbending явления в обычном пространстве были выполнены для четно-четных изотопов Te с четными числами нейтронов от N = 68 до 78, и проведены сравнения с результатами для Гауссова пространства. Описана ядерна структура yrast смуг (зон) парно-парних iзотопiв ¹²⁰⁻¹³⁰Te як наслiдок явища backbending момента iнерцiї. Квадрат обиртової частоти, момент iнерцiї i енергiя Фермi цих ядер були обрахованi iз наявних експериментальних даних. Ми вивчили системний момент iнерцiї i енергiю Фермi, як функцiї парного нейтронного числа ¹²⁰⁻¹³⁰Te iзотопiв. Момент iнерцiї, як функцiя квадрата кутової швидкостi обертання для парних чисел нейтронiв вiд N = 68 до 78 в Te iзотопах, проявляє backbending властивостi. Були вивченi backbending явища для парно-парних iзотопiв Te в звичайному просторi для парних чисел нейтронiв вiд N = 68 до 78, i проведенi порiвняння з даними для Гауссового простору. This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No.(662-016-D1434). 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 Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes Характеризация явления backbending в четно-четных изотопах ¹²⁰⁻¹³⁰Te Характеризацiя явища backbending в парно-парних iзотопах ¹²⁰⁻¹³⁰Te Article published earlier |
| spellingShingle | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes Hussein, M. Hossainc, I. Mansour, S.A. Ядерная физика и элементарные частицы |
| title | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes |
| title_alt | Характеризация явления backbending в четно-четных изотопах ¹²⁰⁻¹³⁰Te Характеризацiя явища backbending в парно-парних iзотопах ¹²⁰⁻¹³⁰Te |
| title_full | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes |
| title_fullStr | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes |
| title_full_unstemmed | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes |
| title_short | Characterization of backbending in even-even ¹²⁰⁻¹³⁰Te isotopes |
| title_sort | characterization of backbending in even-even ¹²⁰⁻¹³⁰te isotopes |
| topic | Ядерная физика и элементарные частицы |
| topic_facet | Ядерная физика и элементарные частицы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/80499 |
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