A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes
In this paper, even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes have been studied the ground state bands using Matlab computer code (IBM-1.Mat). We apply the interacting boson model-1 (IBM-1) formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12⁺ have been...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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irk-123456789-1945582023-11-27T16:14:10Z A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes Hossain, I. Kassim, Huda H. Sharrad, Fadhil I. Al-Jubbori, Mushtaq A. Salam, A. M., Saleem In this paper, even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes have been studied the ground state bands using Matlab computer code (IBM-1.Mat). We apply the interacting boson model-1 (IBM-1) formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12⁺ have been obtained for ¹⁰⁴⁻¹⁰⁶Ru isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for these nuclei. The published experimental and calculated R4/2 values indicate that the even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes have O(6) dynamic symmetry. The present results have been compared to the published experimental data and are found good harmony with each other. The outcome of our investigation of the potential energy surfaces (PES) of both isotopes belonging to O(6) character. Використовуючи Matlab комп’ютерний код (IBM-1.Mat), досліджені полоси основних станів парно-парних ізотопів ¹⁰⁴⁻¹⁰⁶Ru. Ми застосували формулу моделі-1 взаємодіючих бозонів (IBM-1) для O(6) симетрії Ru-ізотопів для N=60, 62 нейтронів. Теоретичні значення енергетичних рівнів до значень спін-парності 12⁺ були отримані для ізотопів ¹⁰⁴⁻¹⁰⁶Ru. Для цих ядер обчислені yrast-стани, γ-смуги, β-смуги та B(E2)-величини. Опубліковані експериментальні та розрахункові величини R4/2 показують, що парно-парні ізотопи ¹⁰⁴⁻¹⁰⁶Ru мають O(6) динамічну симетрію. Ці розрахунки порівнювалися з опублікованими даними експериментальних досліджень, і знайдено добре узгодження між ними. Наші дослідження поверхні потенціальної енергії (ППЕ) вказують на O(6) характер обох ізотопів. Используя Matlab компьютерный код (IBM-1.Mat), исследовались полосы основных состояний четно-четных изотопов ¹⁰⁴⁻¹⁰⁶Ru. Мы применили формулу модели-1 взаимодействующих бозонов (IBM-1) для O(6) симметрии Ru-изотопов для N=60, 62 нейтронов. Теоретические значения энергетических уровней до значений спин-четности 12⁺ были получены для изотопов ¹⁰⁴⁻¹⁰⁶Ru. Для этих ядер вычислены yrast-состояния, γ-полосы, β-полосы и B(E2)-величины. Опубликованные экспериментальные и расчетные величины R4/2 показывают, что четно-четные изотопы ¹⁰⁴⁻¹⁰⁶Ru имеют O(6) динамическую симметрию. Эти результаты сравнивались с опубликованными данными экспериментальных исследований, и найдено хорошее согласие между ними. Наши исследования поверхности потенциальной энергии (ППЭ) указывают на O(6) характер обоих изотопов. 2020 Article A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes / I. Hossain, Huda H. Kassim, Fadhil I. Sharrad, Mushtaq A. Al-Jubbori, A. Salam, M. Saleem // Problems of atomic science and tecnology. — 2020. — № 5. — С. 13-18. — Бібліогр.: 23 назв. — англ. 1562-6016 PACS: 21.10.-k;42.40.Ht; 42.30.Kq http://dspace.nbuv.gov.ua/handle/123456789/194558 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
In this paper, even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes have been studied the ground state bands using Matlab computer code (IBM-1.Mat). We apply the interacting boson model-1 (IBM-1) formula for O(6) symmetry in Ru isotopes with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12⁺ have been obtained for ¹⁰⁴⁻¹⁰⁶Ru isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for these nuclei. The published experimental and calculated R4/2 values indicate that the even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes have O(6) dynamic symmetry. The present results have been compared to the published experimental data and are found good harmony with each other. The outcome of our investigation of the potential energy surfaces (PES) of both isotopes belonging to O(6) character. |
| format |
Article |
| author |
Hossain, I. Kassim, Huda H. Sharrad, Fadhil I. Al-Jubbori, Mushtaq A. Salam, A. M., Saleem |
| spellingShingle |
Hossain, I. Kassim, Huda H. Sharrad, Fadhil I. Al-Jubbori, Mushtaq A. Salam, A. M., Saleem A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes Вопросы атомной науки и техники |
| author_facet |
Hossain, I. Kassim, Huda H. Sharrad, Fadhil I. Al-Jubbori, Mushtaq A. Salam, A. M., Saleem |
| author_sort |
Hossain, I. |
| title |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes |
| title_short |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes |
| title_full |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes |
| title_fullStr |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes |
| title_full_unstemmed |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes |
| title_sort |
study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶ru isotopes |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2020 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/194558 |
| citation_txt |
A study on the ground states structure of even-even ¹⁰⁴⁻¹⁰⁶Ru isotopes / I. Hossain, Huda H. Kassim, Fadhil I. Sharrad, Mushtaq A. Al-Jubbori, A. Salam, M. Saleem // Problems of atomic science and tecnology. — 2020. — № 5. — С. 13-18. — Бібліогр.: 23 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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| fulltext |
A STUDY ON THE GROUND STATES STRUCTURE OF
EVEN-EVEN 104−106Ru ISOTOPES
I.Hossain1∗, HudaH.Kassim2, Fadhil I. Sharrad2, MushtaqA.Al-Jubbori3,
A. Salam4, M. Saleem5
1Department of Physics, Rabigh college of Science and Arts, 21911 Rabigh, King Abdulaziz University,
Jeddah, Saudi Arabia;
2Department of Physics, College of Science, Kerbala University, 56001 Karbala, Iraq;
3Department of Physics, College of Education for Pure Science, University of Mosul, 41001 Mosul, Iraq;
4Department of chemistry, Rabigh college of Science and Arts, 21911 Rabigh, King Abdulaziz University,
Jeddah, Saudi Arabia;
5Department of Industrial Engineering, Faculty of Engineering-Rabigh, King Abdulaziz University,
Saudi Arabia
(Received May 6, 2019)
In this paper, even-even 104−106Ru isotopes have been studied the ground state bands using Matlab computer
code (IBM-1.Mat). We apply the interacting boson model-1 (IBM-1) formula for O(6) symmetry in Ru isotopes
with neutron N = 60, 62. The theoretical energy levels up to spin-parity 12+ have been obtained for 104−106Ru
isotopes. The yrast states, gamma band, beta band, and B(E2) values are calculated for these nuclei. The published
experimental and calculatedR4/2 values indicate that the even-even 104−106Ru isotopes haveO(6) dynamic symmetry.
The present results have been compared to the published experimental data and are found good harmony with each
other. The outcome of our investigation of the potential energy surfaces (PES) of both isotopes belonging to O(6)
character.
PACS: 21.10.-k;42.40.Ht; 42.30.Kq
1. INTRODUCTION
Recently Ruthenium isotope has been a focus of the
nuclear structure of many theoretical and experimen-
tal investigations. The low-lying even nuclei had
been successfully explained nuclear collective charac-
ters using IBM-1[1]. In the first beginning the collec-
tive states can be describes by a system of identical
bosons NB . These are S-boson (L = 0) and d-boson
(L = 2). There is no discrepancy between neutron
and proton in IBM-1. There are three dynamical
symmetries indicated by U(5) analogous to spherical
vibrator, SU(3) deformed rotor and O(6) γ-soft. The
microscopic a harmonic vibrator approach (MAVA)
used in investigating the lower level collective states
in Ruthenium isotopes [2].
The Ruthenium isotopes have atomic number
Z = 44. It belongs near to closed shell Sn (magic
number Z = 50). The external forms of even
104−106Ru isotopes have πg−6
9/2 (6 proton holes) and
νg10,129/2 (10 and 12 neutron particles) close to magic
number 50. This configuration has been investigated
the ground state structure from spherical to deformed
symmetry. The edifice of yrast levels and electromag-
netic strength of Ru isotopes were studied by many
scientists [3-7].
Recently, the properties of the yrast level were
studied in Pd isotopes with even neutron N = 54...64
[8]. The electromagnetic reduced transition strength
of Cd isotopes with N = 66...74 were investigated [9].
The B(E2) value of yrast band of even 102−112Pd and
96−102Ru isotopes were investigated by IBM-1[10,11].
The lower level of 184W and 184Os nuclei were inves-
tigated [12].
The present aim particularly focuses the struc-
ture of the yrast band, gamma band and beta band,
electromagnetic transition and the potential energy
surfaces to find the dynamical symmetry of even
104−106Ru isotopes by the application of IBM.
2. METHODOLOGY
The Interacting Boson Model (IBM) gives occupation
to truncated model space for nuclei with N number of
nucleons. It provides a quantitative description of in-
distinguishable particles with forming pairs of L = 0
and 2. The Hamiltonian of IBM-1[13] is given
H = ΣN
i=1εi +ΣN
i<jVij , (1)
where εi indicates energy of boson, Vij indicates po-
tential energy of boson between i and j. Hamiltonian
∗Corresponding author E-mail address: mihossain@kau.edu.sa
ISSN 1562-6016. PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2020, N5(129).
Series: Nuclear Physics Investigations (74), p.13-18.
13
is given from multi-pole form [14]
Ĥ = εn̂d + α0P̂ .P̂ + α1L̂.L̂+ α2Q̂.Q̂+
+ α3T̂3.T̂3 + α4T̂4.T̂4 , (2)
where
n̂d = (d+, d̃), P̂ =
1
2
(d̃.d̃)− 1
2
(s̃.s̃),
L̂ =
√
10[d+ × d̃](1),
Q̂ = [d+ × s̃+ s+ × d̃](2) −
√
7
2
[d+ × d̃](2),
T̂3 = [d+ × d̃](3), T̂4 = [d+ × d̃](4).
Here P̂ ( the pairing operator), Q̂ (quadrupole oper-
ator), ( number of n̂d boson), L̂ (operator of angu-
lar momentum), and T̂3 (octupole operators), and T̂4
(hexadecapole operators).
The Hamiltonian starting with U(6) and complete
with group O(2) as given in Eq.(2) is bringing to
a lower state of three limits, gamma-soft O(6), vi-
bration U(5) and rotational SU(3) nuclei [15]. It is
known that the limit in the O(6), SU(3) and U(5)
the parameters are α0, α2, and ε respectively.
The Hamiltonian and eigen-values for the three
limits [16]:
U(5) :
ĤU(5) = εn̂d + α1L̂.L̂+ α3T̂3.T̂3 + α4T̂4.T̂4 ,
E(nd, σ, L) = εnd +K1nd(nd + 4) +
+K4σ(σ + 3) +K5L(L+ 1), (3)
with
K1 = 1/12 a1 ,
K4 = −1/10 a1 + 1/7 a3 − 3/70 a4 ,
K5 = −1/14 a3 + 1/14 a4 .
O(6) :
ĤO(6) = α0P̂ .P̂ + α1L̂.L̂+ α3T̂3.T̂3 ,
E(σ, ν, L) = K3 [N(N + 4)− σ(σ + 4)] +
+K4ν(ν + 3) +K5L(L+ 1), (4)
with
K1 = 1/4 a0 ,
K4 = 1/2 a3 ,
K5 = −1/10 a3 + a1 .
SU(3) :
ĤSU(3) = α1L̂.L̂+ α2Q̂.Q̂ ,
E(λ, µ, L) = K2
[
λ2 + µ2 + λµ+ 3(λ+ µ)
]
+
+K5L(L+ 1), (5)
with
K2 = 1/2 a2 ,
K5 = a1 − 3/8 a2 .
K1, K2, K3, K4, and K5 are other strength parame-
ters.
Then applying particular limit of symmetry
(O(6), SU(3), U(5)) to determine the frame of a set
of nuclei is more advantageous than full Hamiltonian
of IBM-1. It comprise multi-free parameters those
make it simple to fit the structure of a nuclei.
3. RESULTS AND DISCUSSION
The obtained results can be discussed for yrast state,
γ-band, β-band, effective charge, transition probabil-
ities B(E2), maxing ratio and potential energy sur-
faces. The calculations were done by IBM-1.
The γ-unstable limit has applied for 104,106Ru nu-
clei using published data of energy ratios (E2 : E4 :
E6 : E8 = 1 : 2.5 : 4.5 : 6.5). In the IBM-1, the even
104,106Ru nuclei have 3 proton boson holes and 5 and
6 neutron boson particle respectively. Therefore to-
tal bosons numbers of 104Ru and 106Ru nuclei are
8 and 9, respectively. The IBM-1 models carry out
with no difference between the bosons of proton and
neutron. The energy ratio R = E4+1 /E2+1 gives the
information of the symmetry shapes of a nucleus. The
symbol E2+1 and E4+1 are at energy level 2+1 and 4+1
respectively. It is known that the R = E4+1 /E2+1 ≈ 2
is for U(5), R = E4+1 /E2+1 ≈ 2.5 is for O(6) and
R = E4+1 /E2+1 ≈ 3.33 for SU(3) [17,18]. The experi-
mental R4/2 of 104Ru and 106Ru isotopes is 2.48 and
2.60, respectively. In Fig.1 shows R4/2 values and
found O(6) symmetry in 104Ru and 106Ru isotopes.
1.5
2
2.5
3
3.5
4
E
4
/E
2
v
a
lu
e
Neutron number
U(5)
SU(3)
O(6)
60 62
Fig.1. The arrows indicate the U(5), O(6) and
SU(3) limits. The E(4−1 )/E(2−1 ) values of ex-
perimental data [22] of the 104,106Ru isotopes are
presented as function of neutrons
The best fit was taken up to 12+ of Ru isotopes
with neutron N = 60, 62. The parameters were de-
termined the experimental eigen values (E(nd, σ, L))
from the equation (4). Where nd, σ, and L are quan-
tum numbers. The parameters are shown in Table 1.
14
Table 1. Adopted values of parameters used for IBM-1 calculations. All parameters are given in MeV ,
excepted N and CHQ
A N ε a0 a1 a2 a3 a4 CHQ(χ)
104Ru 8 0.000 0.1098 0.0180 0.000 0.1770 0.000 0.000
106Ru 9 0.000 0.0990 0.0102 0.000 0.1513 0.000 0.000
The calculated energy levels as well as experimen-
tal data are presented in Table 2. According to the
weight of fitting the Ru − 104 and Ru − 106 nuclei
are good candidates of O(6) symmetry.
Table 2. g-band (in MeV ) for even 104−106Ru
nuclei
IBM Exp. IBM Exp.
Jπ
104Ru 106Ru
0+ 0.000 0.000 0.000 0.000
2+ 0.3558 0.3580 0.2726 0.2700
4+ 0.8910 0.8884 0.6570 0.7147∗
6+ 1.6056 1.5564 1.1532 1.2958∗
8+ 2.4996 2.3204 1.7612 1.9734∗
10+ 3.5730 3.1119 2.4810 2.7050
12+ 4.8258 — 3.3126 3.4500∗
The present data of γ-bands and β-bands are pre-
sented in Tables 3 and 4 and they are compared with
published measured data. From the tables, the IBM
calculations and experimental results are in good
agreements [19].
Table 3. γ-band (in MeV ) for even 104−106Ru
nuclei
IBM Exp. IBM Exp.
Jπ
104Ru 106Ru
2+ 0.8868 0.8931 0.7256 0.7923
3+ 1.5966 1.2424 1.2610 1.0915∗
4+ 1.5990 1.5026 1.3002 1.3068∗
5+ 2.4870 1.8723∗ 1.9082 1.6411∗
6+ 2.4906 2.1966∗ 1.9670 1.9078∗
7+ 3.5568 —- 2.6672 2.2841∗
8+ 3.4878 —- 2.7456 2.9600∗
9+ 4.8060 — 3.6360 —
Table 4. β-band (in MeV ) for even 104−106Ru
nuclei
IBM Exp. IBM Exp.
Jπ
104Ru 106Ru
0+ 0.9882 0.9882 0.9900 0.9906
2+ 1.3440 1.5154 1.2626 1.3922
4+ 1.8792 2.0808 1.6470 —-
6+ 2.5938 — 2.1432 —
8+ 3.5616 — 2.7512 —
10+ 4.8120 — 3.5380 —
The reduced electric transition probabilities give
the more properties on the structure of nuclei. It is
known that the E2 transition operator must be a Her-
mitian tensor of rank two; consequently, the number
of bosons must be conserved.
TE2 = α2
[
d+s+ s+d
](2)
+ β2
[
d+d
](2)
, (6)
where (s+, d+) are creation and (s, d) are annihilation
operators for bosons s and d. TE2 is the operator of
reduced matrix elements of the E2. α2 is indicated
the effective quadrupole charge and β2 is dimension-
less coefficient, β2 = χα2.
B(E2, Ji → Jf ) =
1
2Ji + 1
∣∣⟨Jf ∥∥TE2
∥∥ Ji⟩∣∣2 . (7)
The parameters, α2 and β2 of Eq.(6), ac-
commodated suitably a set to reproduce the
published B(E2; 2+1 → 0+1 ). The effective
charge (eB) in present calculation is shown in
Table 5. The values of eB were estimated
to reproduce experimentally (B(E2; 2+1 → 0+1 )).
Table 5. Effective charge for even 104−106Ru nuclei
A N eB(eb)
104Ru 8 0.0935
106Ru 9 0.0916
15
The values β2 = 0 for 104,106Ru isotopes because
these nuclei have the O(6) property. The values of
B(E2) data are presented in Table 6 for Ru iso-
topes with neutron N = 60, 62 in this study [19].
The calculated data of IBM-1 is good agreements
with the available published experimental results.
Table 6. The B(E2) value (in e2b2) of even
104−106Ru isotopes
IBM Exp. IBM Exp.
Ji → Jf
104Ru 106Ru
2+1 → 0+1 0.1679 0.1682 0.1966 0.1966
4+1 → 2+1 0.2273 0.2149 0.2689 –
4+2 → 2+2 0.1282 – 0.1541 –
6+1 → 4+1 0.2448 – 0.2941 –
6+2 → 4+2 0.1626 – 0.2000 –
8+1 → 6+1 0.2384 – 0.2934 –
10+1 → 8+1 0.2152 – 0.2747 –
10+2 → 8+2 0.0941 – 0.0000 –
2+2 → 2+1 0.2273 0.1957 0.2689 –
4+2 → 4+1 0.1166 – 0.1401 –
6+2 → 6+1 0.0759 – 0.0933 –
8+2 → 8+1 0.0000 – 0.0659 –
The application of potential energy surface (PES)
gives the information to find microscopic and geomet-
ric shapes such as spherical, prolate, oblate and
γ-independent (γ-soft)). It gives us about symmetry,
the shape of nuclei, the minimum deepness and the
change of the shape. The PES plots using Skyrme
mean field method was drawn by IBM Hamiltonian
[20-23]. The state is a product of the boson creation
operators b+c with
|N, β, γ⟩ = 1√
N !
(
b+c
)N |0⟩ , (8)
b+c = (1 + β2)−12
{
s+ + β
[
cos γ(d+0 )+
+
√
1/2 sin γ(d+2 + d+−2)
]}
. (9)
The energy surface as a function of β and γ, has
been given [1]
E(N, β, γ) =
=
Nεdβ
2
(1 + β2) + N(N−1)
(1+β2)2(α1β4+α2β3 cos 3γ+α3β2+α4)
.
(10)
Here the αi’s are related to the Casimir coefficients
CL, ν2, ν0, u2 and u0. β were indicated the total de-
formation of a nucleus. Then, we can re-write equa-
tion (10) for O(6) limit [19]. Fig.2 shows the contour
plots in the γ− β plane resulting from E(N, β, γ) for
104Ru and 106Ru isotopes. The color lines show the
values of the potential energy surface in MeV . The
potential surfaces are approximately independent of
gamma only. The mapped IBM energy surfaces of
104Ru and 106Ru are O(6) character.
Fig.2. Contour map of potential energy surfaces for Ru isotopes with N = 60, 62
4. CONCLUSIONS
The yrast band, gamma band and beta band, elec-
tromagnetic transition and potential energy surface
of 104Ru and 106Ru isotopes have been calculated
in term of O(6) limit of IBM-1. The levels up to
12+ of 104,106Ru nucleus are found by the Hamilto-
nian of IBM-1. The analyses of the IBM-1 results
suggest a satisfactory agreement with the published
experimental data. It is established that interacting
boson approximations for 104Ru and 106Ru isotopes
are considered as gamma soft O(6) symmetry.
16
ACKNOWLEDGMENTS
This project was funded by the Deanship of Scientific
Research (DSR) at King Abdulaziz University, Jed-
dah, under grant No. G-223-662-37. The authors,
therefore, acknowledge with thanks DSR for techni-
cal and financial support.
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17
ÈÑÑËÅÄÎÂÀÍÈÅ ÑÒÐÓÊÒÓÐÛ ÎÑÍÎÂÍÛÕ ÑÎÑÒÎßÍÈÉ ×ÅÒÍÎ-×ÅÒÍÛÕ
ÈÇÎÒÎÏÎÂ 104−106Ru
I. Hossain, HudaH.Kassim, Fadhil I. Sharrad, MushtaqA.Al-Jubbori, A. Salam, M.
Saleem
Èñïîëüçóÿ Matlab êîìïüþòåðíûé êîä (IBM-1.Mat), èññëåäîâàëèñü ïîëîñû îñíîâíûõ ñîñòîÿíèé ÷åòíî-
÷åòíûõ èçîòîïîâ 104−106Ru. Ìû ïðèìåíèëè ôîðìóëó ìîäåëè-1 âçàèìîäåéñòâóþùèõ áîçîíîâ (IBM-1)
äëÿ O(6) ñèììåòðèè Ru-èçîòîïîâ äëÿ N = 60, 62 íåéòðîíîâ. Òåîðåòè÷åñêèå çíà÷åíèÿ ýíåðãåòè÷åñêèõ
óðîâíåé äî çíà÷åíèé ñïèí-÷åòíîñòè 12+ áûëè ïîëó÷åíû äëÿ èçîòîïîâ 104−106Ru. Äëÿ ýòèõ ÿäåð âû÷èñ-
ëåíû yrast-ñîñòîÿíèÿ, γ-ïîëîñû, β-ïîëîñû è B(E2)-âåëè÷èíû. Îïóáëèêîâàííûå ýêñïåðèìåíòàëüíûå
è ðàñ÷åòíûå âåëè÷èíû R4/2 ïîêàçûâàþò, ÷òî ÷åòíî-÷åòíûå èçîòîïû 104−106Ru èìåþò O(6) äèíàìè-
÷åñêóþ ñèììåòðèþ. Ýòè ðåçóëüòàòû ñðàâíèâàëèñü ñ îïóáëèêîâàííûìè äàííûìè ýêñïåðèìåíòàëüíûõ
èññëåäîâàíèé, è íàéäåíî õîðîøåå ñîãëàñèå ìåæäó íèìè. Íàøè èññëåäîâàíèÿ ïîâåðõíîñòè ïîòåíöèàëü-
íîé ýíåðãèè (ÏÏÝ) óêàçûâàþò íà O(6) õàðàêòåð îáîèõ èçîòîïîâ.
ÄÎÑËIÄÆÅÍÍß ÑÒÐÓÊÒÓÐÈ ÃÎËÎÂÍÈÕ ÑÒÀÍIÂ ÏÀÐÍÎ-ÏÀÐÍÈÕ IÇÎÒÎÏIÂ
104−106Ru
I. Hossain, HudaH.Kassim, Fadhil I. Sharrad, MushtaqA.Al-Jubbori, A. Salam, M. Saleem
Âèêîðèñòîâóþ÷è Matlab êîìï'þòåðíèé êîä (IBM-1.Mat), äîñëiäæåíi ïîëîñè îñíîâíèõ ñòàíiâ ïàðíî-
ïàðíèõ içîòîïiâ 104−106Ru. Ìè çàñòîñóâàëè ôîðìóëó ìîäåëi-1 âçà¹ìîäiþ÷èõ áîçîíiâ (IBM-1) äëÿ O(6)
ñèìåòði¨ Ru-içîòîïiâ äëÿ N = 60, 62 íåéòðîíiâ. Òåîðåòè÷íi çíà÷åííÿ åíåðãåòè÷íèõ ðiâíiâ äî çíà÷åíü
ñïií-ïàðíîñòi 12+ áóëè îòðèìàíi äëÿ içîòîïiâ 104−106Ru. Äëÿ öèõ ÿäåð îá÷èñëåíi yrast-ñòàíè, γ-ñìóãè,
β-ñìóãè òà B(E2)-âåëè÷èíè. Îïóáëiêîâàíi åêñïåðèìåíòàëüíi òà ðîçðàõóíêîâi âåëè÷èíè R4/2 ïîêàçó-
þòü, ùî ïàðíî-ïàðíi içîòîïè 104−106Ru ìàþòü O(6) äèíàìi÷íó ñèìåòðiþ. Öi ðîçðàõóíêè ïîðiâíþâàëèñü
ç îïóáëiêîâàíèìè äàíèìè åêñïåðèìåíòàëüíèõ äîñëiäæåíü, i çíàéäåíî äîáðå óçãîäæåííÿ ìiæ íèìè. Íà-
øi äîñëiäæåííÿ ïîâåðõíi ïîòåíöiàëüíî¨ åíåðãi¨ (ÏÏÅ) âêàçóþò íà O(6) õàðàêòåð îáîõ içîòîïiâ.
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