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153
NEGATIVE PARITY BANDS IN 236U
Ne’matjonov Shuhratjon Rustamjon ugli,
Nishonov Azizbek Nazirjon ugli.
Namangan state technical university, Namangan, Uzbekistan
shuxrat.nematjonov1995@gmail.com.
The energy characteristics of the octupole states of the 236U nucleus are studied within the
framework of a phenomenological model that takes into account the Coriolis mixing of the states
of rotational bands. The mixing of states of low-lying bands of negative parity is considered. The
energies and structure of states of rotational bands with
р
K
=
0
and 1
are calculated.
Theoretical energy values are compared with experimental data, which gives good agreement.
Keywords:
energy, nucleus, spin, negative parity, Coriolis mixing, rotational band.
Introduction
Currently, in the physics of the structure of the atomic nucleus, experimental and theoretical
studies of states of negative parity of nuclei in the actinide region are very relevant [1]. The
collective nature of low-lying states of negative parity in actinides was clarified in calculations
made within the framework of the random phase method [2-4]. These predictions are consistent
with the Coulomb excitation data. In the 236U nucleus, rotational bands are known, constructed
in the ground and octupole-vibrational states. The main experiment for studying the properties of
excited states is the study of Coulomb excitation in reactions with heavy ions [5-7]. The
probabilities of transitions from excited levels to states of the ground band are associated with
significant violations of the Alag rules.
Experimental data were analyzed within the framework of microscopic [7] and
phenomenological models [8]. The analyzed data within the framework of the microscopic
model [7] have discrepancies in comparison with experiment.
The lowest band with a head energy of E0=688 keV with a base of
р
0
K
=
is traced to spin
21
I
=
h
. In the band with
р
1
K
=
with a head energy of E1=967 keV, levels with even spins up
to
4
I
=
h
and with odd spins up to
5
I
=
h
are known [1].
In this work, to study the properties of negative parity states in 236U, we use a
phenomenological model [9, 10], which considers the Coriolis mixing of the states of the bands
with
р
0
K
=
and
р
1
K
=
. The energy spectrum and the structure of states of low-lying octupole
bands are described.
Model
Within the framework of the model [9, 10], the Hamiltonian of the kernel has the following form:
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154
2
( )
( )
rot
KK
H H
I
H
I
=
+
,
(1)
Where
01
1
.
1
( )
1 ( 1)
( )( )
2
I
KK
K KK
rot
x
KK
H
I
I j
±
= w d
w
d
(2)
Here
w
0 and
w
1 are the head energies of the bases of the rotational bands from
р
0
K
=
and
р
1
K
=
, respectively;
р
1
K
=
– angular frequency of rotation of the core;
01
€
( )
0
1
x
x
j
j
=
–
matrix element between internal wave functions.
We look for the wave function in the form:
{
}
2
,0
,
,
ш
2
1
16р
1 д
(и)
( 1)
(и)
0
I
K K
K
K
I
I K
I
M K
K
M K
K
I
IMK
D
b
D
b
+
+
+
+
=
+
+
,
(3)
where
ш
I
K K
is the mixing coefficient of the states of rotational bands;
,
I
M K
D
– Wigner function;
K
b
+
- single-phonon states that serve as the bases of negative parity bands:
3
0
0
K
K
b
b
l
+
+
=
=
with
р
1
K
=
and
р
1
K
=
Solving the Schrödinger equation
(
( )
( ))ш
0
I
K v
v
K v
H
I
I
e
=
,
(4)
determine the eigenvalues of the energy states of
р
0
K
=
and
р
1
K
=
bands:
2
2
2
0
1
0,1
0
1
0,1
(щ -щ )
4щ ( )( )
щ +щ
е( )
2
4
rot
x
I j
I
+
=
±
. (5)
The total energy of the state is determined by the formula
( )
( )
( )
v
rot
v
E I
E I
I
=
+ e
.
(6)
The energy of the rotating core Erot(I) is determined using the Harris parameterization [13]:
2
4
0
1
1
3
( )
( )
( )
2
4
rot
rot
rot
E I
I
I
=
w
+
w
,
(7)
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2
3
0
1
( 1)
( )
( )
rot
rot
I I
I
I
+ = w
+ w
,
(8)
where
0 and
1 are the inertial parameters of the rotating frame
Having solved the cubic equation (8), we determine the rotation frequency of the frame
w
rot(I).
Equation (8) has one real and two complex solutions. The actual solution to equation (8) is the
value of the rotation frequency
w
rot(I), which is determined by the following analytical formula
1/3
1/2
3
0
2
1
1
1
1/3
1/2
3
0
2
1
1
1
(
1)
(
1)
( )
2
3
4
(
1)
(
1)
.
2
3
4
rot
I I
I I
I
I I
I I
+
+
w
=
+
+
+
+
+
+
+
(9)
For the eigenwave functions of states of negative parity, taking into account the Coriolis
interaction of the states of
р
0
K
=
and
р
1
K
=
bands, we have the following formula
1
2
,
,
,
0
ш
I
I
I
K K
K K
K v
v
=
= F
F
,
(10)
where
00
11
01
0
1
10
01
1
0
01
1
= (щ
е ( )),
щ ( )( ) ,
2
1
=щ е ( ),
=
щ ( )( ) ,
2
I
I
rot
x
I
I
rot
x
Ф
I
Ф
I j
Ф
I
Ф
I j
=
, (11)
here
and
K
K
take the values 0 and 1
Numerical calculations
Calculations were carried out for the 236U core. We considered the mixing of states
р
0
K
=
and
р
1
K
=
rotational bands. In describing the energy of states, the model parameters are the head
energies
w
0,
w
1 and the matrix element (jx)01, which describes the Coriolis mixing of rotational
bands, the values of which were determined using the least squares method from the condition of
best agreement between the calculated and experimental energies. In works [12-15], within the
framework of this model, we studied the properties of rotational states of positive parity of nuclei
of the rare-earth region, where the inertial parameters of the rotating core
0 and
1 were
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determined by the two-parameter Harris formula [11], using experimental ground band energies
up to spin
8
I
h
In this case, to obtain good agreement between the energies of the states of the octupole bands,
we had to vary the parameters
0 and
1, determined by the method described above. The
values of the model parameters used in the calculations are given in Table. 1.
Table 1. Model parameter values used.
w
0, МeV
w
1, МeV
(jx)01
0V
2
/
М eV
h
1
4
3
/
МeV
h
0.679
0.948
2.34
66.93
385.7981
Note:
w
K -head energies of bands from bands
р
0
K
=
and
р
1
K
=
; (jx)01
matrix element of
Coriolis mixing;
0 and
1
are the inertial parameters of the rotating frame
In table 2 shows the calculated values of the angular frequency
w
rot(I) of the rotational Erot(I)
and internal
e
K(I) energies, as well as the theoretical and experimental energies of the rotational
states of the bands with
р
0
K
=
and
р
1
K
=
.
Table 2. Energy characteristics of the excited states of bands with
0
K
p
=
and
1
K
p
=
I
p
w
rot(I)
Erot(I)
e
0(I)
E0(I)
e
1(I)
E1(I)
Exp. [1] Our work
Exp. [1]
Our work
1
0.021
0.015
0.675
0.688
0.690
0.952
(0.967)
0.967
2
0.037
0.045
0.988
0.993
3
0.051
0.089
0.655
0.744
0.744
0.972
1.037
1.061
4
0.066
0.149
1.070
1.096
5
0.079
0.220
0.626
0.848
0.846
1.001
(1.164)
1.221
6
0.093
0.310
1.254
7
0.105
0.405
0.594
0.999
0.999
1.033
1.438
8
0.119
0.523
1.464
9
0.129
0.640
0.561
1.199
1.201
1.066
1.706
10
0.142
0.784
1.723
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11
0.152
0.922
0.529
1.444
1.451
1.098
12
0.164
1.090
13
0.172
1.246
0.499
1.733
1.745
1.129
14
0.183
1.437
15
0.191
1.610
0.470
2.061
2.079
1.152
16
0.202
1.823
17
0.209
2.010
0.440
2.423
2.453
1.187
18
0.219
2.243
19
0.225
2.444
0.417
2.823
2.861
1.212
20
0.235
2.697
21
0.241
2.911
0.393
3.249
3.304
1.236
22
0.250
3.182
23
0.255
3.407
0.368
3.778
1.259
Figure 1 compares the theoretical and experimental energies of the states of the negative parity
bands with
р
0
K
=
and
р
1
K
=
. As can be seen, the model used [9, 10] reproduces well the
experimental data on the energy of the band with
р
0
K
=
. There is a discrepancy between the
experimental and theoretical values in the energies of the
р
1
K
=
band states.
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0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
236
U
(
)
1
odd
K
p
=
(
)
1
even
K
p
=
0
K
p
=
,
x
E MeV
8
6
4
2
9
7
3
5
1
23
21
19
17
15
13
11
9
7
5
31
Fig. 1. Comparison of theoretical and experimental energies of states of negative parity bands
with
р
0
K
=
and
р
1
K
=
0.00
0.01
0.02
0.03
0.04
0.05
0.06
130
140
150
160
170
180
190
236
U
core
(
0 )
theor
eff
K
p
=
(
0 )
exp
eff
K
p
=
2
2
( ),
I
МeV
w
1
2 ( ),
I
М eV
Fig. 2. Dependence of the moment of inertia on the square of the rotation frequency for octupole
bands 236U:
exp
( )
eff
K
and
( )
theor
eff
K
- effective values of experimental and theoretical moments
of inertia
р
0
K
=
;
core
– moment of inertia of the rotating frame.
Figure 2 shows the effective values of the experimental and theoretical moments of inertia
р
0
K
=
, as well as the moment of inertia of the rotating frame
core
. Experimental values of
the effective moments of inertia of
р
0
K
=
bands are always greater than
core
. Considering the
mixing of states band with
р
0
K
=
increases the effective values of the moments of inertia of
band
р
0
K
=
.
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It should be noted that in the scheme used, states with even spins do not participate in Coriolis
mixing; the energies of levels with even spins are determined by the following formula:
1
1
( )
( )
ro t
E
I
E
I
= w +
.
Figure 3 shows the spin dependence of the internal energy of states of
р
0
K
=
and
р
1
K
=
bands. As the angular momentum I increases, the angular frequency
w
rot(I). increases. Therefore,
the calculation is based on the Coriolis interaction between the bands. Therefore, as I increases,
the internal energies of the states of the
р
0
K
=
and
р
1
K
=
bands repel each other more
strongly.
Figure 4 shows the mixing amplitude coefficients
р
0
K
=
0
I
K
y
for the states of the
р
0
K
=
band depending on the spin. As I increases, component
00
I
y
decreases and component
01
I
y
increases. At large spin values, this effect should manifest itself in nonadiabatic E1-transitions.
1
3
5
7
9
11
13
15
17
19
21
23
25
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1
K
=
0
K
=
( )
K
I
e
236
U
,
I
h
Fig. 3. Spin dependence of internal energy
e
K(I) (for states with odd values of I).
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1
3
5
7
9
11
13
15
17
19
21
0.0
0.2
0.4
0.6
0.8
1.0
,
I
h
236
U
01
y
00
y
0
( )
K
I
y
Fig. 4. State structure of
р
0
K
=
bands.
Conclusion
The energy properties of the states of negative parity of the 236U nucleus have been studied
within the framework of a phenomenological model, which considers the Coriolis mixing of the
states of low-lying rotational bands with
р
0
K
=
and
р
1
K
=
. The energies and structure of
states of octupole rotational bands are calculated. The calculated energy values are compared
with the available experimental data. The model used reproduces well the experimental data on
the energy of the
р
0
K
=
band. But in the energies of states with even spins of the
р
1
K
=
band,
a discrepancy is observed between the experimental and theoretical values, which increases with
increasing angular momentum I. It is shown that to improve the description of the energy of the
states of the
р
1
K
=
band, it is also necessary to take into account the mixing of the states of the
р
2
K
=
and
р
3
K
=
bands
Литература
(References)
[1] Sh. Zhu, “Nuclear Data Sheets for A=236”, Nucl. Data Sheets, vol.182, pp. 2-129, 2022
[2] V.G. Soloviev, P. Vogel. Phys. Lett. 6, 126-128 (1963). https://doi.org/10.1016/0031-
9163(63)90248-8
[3] V.G. Soloviev, P. Fogel, А.А. Korneychuk. Izvestiya AN SSSR, seriya fizicheskaya 28,
1599-1616 (1964).
[4] А. Faessler, A. Plastino. Zeitschrift für Physik, 203, 333-345 (1967).
https://doi.org/10.1007/BF01331060.
[5] K. Abu Saleem, R.V. Janssens, M.P. Carpenter, F.G. Kondev, I. Weidenhover, I. Ahmad,
J. Caggiano, P. Chowdhury, J.A. Cizewski, D. Cline, V. Devlin, N. Fotiades, J.P. Greene, G.
Hackman, A. Heinz, T.L. Khoo, T. Lauritsen, C.J. Lister, A. O. Macchiavelli, E.H. Seabury, D.
Volume 15 Issue 07, July 2025
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http://www.internationaljournal.co.in/index.php/jasass
161
Seweryniak,
A.
Sonzogni,
C.Y.
Wu.
Phys.
Rev,
C
70,
024310
(2004).
https://doi.org/10.1103/PhysRevC.70.024310.
[6] S. Zhu, R.V. Janssens, G.J. Lane, I. Wiedenhower, M.P. Carpenter, I. Ahmad, A.P. Byrne,
P. Chowdhury, D. Cline, A.N. Deacon, G.D. Dracoulis, S.J. Freeman, N.J. Hammond, G.D.
Jones, T.L. Khoo, F.G. Kondev, T. Lauretsin, C.J. Lister, A.O. Macchiavelli, E.F. Moore, D.
Seweryniak, J.F. Smith, C.Y. Wu.
Phys. Lett. B 618, 51-59 (2005).
doi:10.1016/j.physletb.2005.05.039.
[7] M. Bender, P.H. Heenen and P. Bonche. Phys. Rev. С 70, 054304 (2004).
https://doi.org/10.1103/PhysRevC.70.054304.
[8]
I.N. Mikhaylov, R.Kh. Safarov, P.N. Usmanov, Sh. Brianson. Yadernaya Fizika 38,
No.2(8), 297-307 (1983).
[9]
P.N. Usmanov, A.A. Solnyshkin, A.I. Vdovin, U.S. Salikhbaev, Phys. of Atomic Nuclei
77, No.11, 1343-1349 (2014). https://doi.org/10.1134/S1063778814100135; 1408-1414 (2014).
https://doi.org/10.7868/S0044002714100146.
[10] K.Ya. Gromov, P.N. Usmanov, A.Kh. Kholmatov, Т.А. Islamov, Yu.S. Butabaev, R.A.
Niyazov. Izvestiya RAN,
seriya fizicheskaya 56, No.11, 36-42 (1992).
[11] S.M.
Harries.
Phys.
Rev.
138,
509-513
(1965).
https://doi.org/10.1103/PhysRev.138.B509.
[12] P.N. Usmanov, A.A. Okhunov, U.S. Salikhbaev, A.I. Vdovin. Physics of Particles and
Nuclei Letters 7, No.3, 185-191 (2010). https://doi.org/10.1134/S1547477110030040.
[13] A.A. Okhunov, P.N. Usmanov, Torla Hj. Hassan, M.M. Abu El Sheikh. Available online
www.sciencedirect.com, Science direct. Materials Today: Proceedings 17, 424-429 (2019).
https://doi.org/10.1016/J.MATPR.2019.06.468
[14] P.N. Usmanov, E.K. Yusupov. IIUM Engineering Journal 21, No.1, 167-174 (2021).
https://doi.org/10.31436/iiumej.v22i1.1497.
[15] P.N. Usmanov, A.I. Vdovin, E.K. Yusupov, U.S. Salikhbaev. Phys. Part. Nucl. Letters 16,
No.6, 706-712 (2019). DOI:10.1134/S1547477119060530
