APPLICATION OF DIGITAL CELLULAR RADIO FOR MOBILE LOCATION ESTIMATION


IIUM Engineering Journal, Vol. 22, No. 1, 2021 Usmanov and Yusupov 
https://doi.org/10.31436/iiumej.v22i1.1497 

ENERGY AND STRUCTURE STATES OF LOW-LYING 

BANDS IN 156Gd 

PAZLITDIN NURITDINOVICH USMANOV* AND 

ELMUROD KUCHKARBOYEVICH YUSUPOV 

Department of Physics, Namangan Engineering and Technology Institute, 

160115 Namangan, Uzbekistan 

*
Corresponding author: usmanov1956.56@mail.ru 

      (Received: 3rd June 2020; Accepted: 3rd September 2020; Published on-line: 4th January 2021) 

ABSTRACT:  The experimental results of the literary and electronic nuclear database for 
156Gd were summarized and analyzed. Inertial parameters of rotating core were determined 

using the Harris method. The theoretical values of energy and wave functions were 

calculated within the framework of a phenomenological model that takes into account 

Coriolis mixing of state rotational bands. The calculated energy values were compared 

with existing experimental data, which were in good agreement.  

ABSTRAK: Hasil dapatan kajian melalui eksperimen pangkalan data nuklear dan 

elektronik bagi 156Gd diringkaskan dan dianalisis. Parameter inersia putaran berputar 

ditentukan menggunakan kaedah Harris. Nilai teori fungsi tenaga dan gelombang dikira 

dalam kerangka model fenomenologi yang mengambil kira campuran Coriolis pada band 

putaran keadaan. Nilai pengiraan tenaga dibandingkan dengan data eksperimen 

memberikan persetujuan yang baik. 

KEYWORDS: nuclei; energy; states; rotational bands; phenomenological model 

1. INTRODUCTION

The gadolinium isotope with a mass A=156 is one of the most studied nuclei. The main

reason is the large cross section ( , γ)n - reaction in 
156

Gd , which provides big opportunities 

for studying the emission spectra in this reaction. Full results on this nucleus are given in 

references [1,2]. Several other nuclear processes supplement information on the levels and 

rotation bands in 
156

Gd . In the reaction of (α,2 )n  data were obtained on the states of 

rotational bands with 
1

0K
 +
= – to 26I

+
= , 

2
0K

 +
= – to 14I

+
= , 

3
0K

 +
= – to 10I

+
= , 

4
0K

 +
= – to 6I

+
= , 

5
0K

 +
= – to 4I

+
= , 

1
2K

 +
= – to 15I

+
= and 

2
2K

 +
= – to 4I

+
= . 

A wealth of experimental information was obtained as a result of studying the reaction 

( , γ)n n  [3]. Based on the totality of the experimental data, one can think that for 
156

Gd  all 

or almost all levels were found up to excitation energy of 2 MeV. In 
156

Gd  five rotational 

bands with 
π

0K
+

=  are known. Therefore, systematic study of the properties of these levels 

is very important to search the corresponding levels in the neighboring nuclei. 

156
Gd and 

158
Gd are the first nuclei in which a new type of excitation was discovered 

– collective states with 
π

1I
+

=  [4]. At present, there have been several collective states 

experimentally observed in 
156

Gd with
π

1I
+

= . Their excitation energies and reduced 

probabilities ( 1)B M   are determined [5]. 

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Nonadiabaticity is observed in the energies and especially in the electromagnetic 

characteristics of the excited states of rotational bands with 
π

0K
+

=  and 
π

2K
+

=  of these 

nuclei [6-9]. In the present work, to explain these nonadiabatics, we use a phenomenological 

model that takes into account Coriolis mixing of low–lying rotational bands [10, 11]. The 

energy levels and wave functions of the rotational band states are calculated. The 

nonadiabatics observed in the energies and wave functions of rotational states are discussed. 

More details of the phenomenological model used are given in Ref. [10] and this model 

was also successfully applied to study the mixing of state bands of both positive and 

negative parity in [11-14]. 

2.   DESCRIPTION OF THE MODEL  

According to the phenomenological model of the nucleus [10], the Hamilton operator 

has the following appearance: 

2

rot ,
( )

K K
H H I H = +  (1) 

, ' , ' rot , ' 1
ˆω δ ω ( ) χ( , )

K K K K K x K K
H I K j K I K 


= −     (2) 

rot rot
ω ( ) ( )I dE I dI= – angular frequency of rotation of the core; 

К
 −  the head energies of 

the rotational bands; ˆ
x

K j K   – matrix element of Coriolis interaction between the 

states of rotational bands; χ( , )I K  – the coefficients are as follows: 

χ( ,0) 1I = ,  

1 2

2
χ( ,1) 1

( 1)
I

I I
= −

+
. 

The Eigen wave function of the Hamilton operator (1) has the following appearance: 


2

',

, ' ' . '

' ',0

2 1
2 (θ)

gr, , 0
16π

(θ) ( 1) (θ) 0 ,
1 δ

I

K K I I K I

M K K M K K

K K

I I I
IMK D

K M

D b D b
+ + +

− −

+
=  +

 
 + + −  

+ 


 (3) 

here 
, '

(θ)
I

M K
D

 
are generalized spherical functions; 0

 
is the vacuum for the 

K
b
+

 
operators 

(in other words, the ground state of the nucleus in the intrinsic reference system); 
,

I

K K
 are 

the amplitudes of mixing states of different bands with the same angular momentum I due 
to the Coriolis interaction. 

Eigen energies and wave functions of the rotational states are determined by 

diagonalization of the Hamiltonian operator (2): 

σ

, ' , ,

I I I

K K K n n K n
H  =   (4) 

The total energies of the states can be found by the following formula: 

rot
( ) ( ) ( )

K K
E I E I I= +

 
(5) 

There are different methods for determining the energy of the core rotational motion of 

the nucleus, for example, Harris proposed to determine the following two-parameter 

formula [15]: 

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3

0 rot 1 rot
( 1) ω ( ) ω ( )I I I I+ =  +   (6) 

2 4

rot 0 rot 1 rot

1 3
( ) ω ( ) ω ( )

2 4
E I I I=  + 

, 

here 
0

  and 
1


 
– inertia parameters of the rotational core. 

3.   CALCULATION AND RESULTS 

According to Bohr–Mottelson and Bengtsson–Frauendorf, at small values of the spin 

of the nucleus, the rotational energy of the nucleus corresponds to the ground rotational 

states energy [16,17]. Therefore, the inertial parameters of the core were determined by the 

Harris method using the experimental energy states of ground band up to spin 8I  . 

Table 1: Values of the model parameters used in calculating level energies 

0  1  12
ω  

22
ω  +ˆ0 11 νjx   

+
ˆ0 12 νjx

   
+

ˆ0 13 νjx
   

+
ˆ0 14 νjx

   
+

ˆ0 15 νjx
   

+
ˆ2 11 νjx

+
   

+
ˆ2 12 νjx

+
   

33.3 169.47 1.076 1.739 0.416 0.45 0.25 0.91 0.42 0.40 0.1 

NOTE: 
2

0
( /MeV) ,

4 3

1
( /MeV )  are the inertial parameters of the rotating core (Harris parameters), 

1,22
ω  (MeV) are energies of the 

π

1,2
2K
+

= band heads, and ˆ
x

K j K   (MeV) are the matrix elements 

of the Coriolis interaction. 

Figure 1 shows the values of the moment of inertia obtained by the Harris method, 

which are compared with the effective values obtained using the following formula 

эфф

эфф

( 1)
( )

ω ( )

I I
I

I

+
 =

, here 

экс экс

эфф

( 1) ( 1)
ω ( )

2

E I E I
I

+ − −
=

 (7) 

It can be seen from the comparison that there is a difference between them at high values 

of angular rotation frequency. I.e. at the large values of the total angular momentum I , 
deviations from the adiabatic theory are observed in the energies of the ground band. Such 

nonadiabaticity is more pronounced in the energies of the vibration bands with 0K
 +
=  and 

2K
 +
= . To determine the causes of nonadiabaticity in the energies of the states of low-

lying rotational bands, we use the phenomenological model presented above. 

The energy of the rotational motion of the nucleus, for the bands included in the basis 

of the Hamiltonian operator (1), is considered the same and its values are determined by 

formula (6). The rotational motion at low frequencies is not related to its internal structure, 

i.e., the rotation of the nucleus does not affect the Hamiltonian basic model (1). Ground, 

five beta band ( 0K
 +
= ), two gamma 2K

 +
=  and fifteen 1K





+
=  rotational bands are 

introduced in the Hamiltonian model. The Schrödinger equation of size (23X23) was solved. 

 

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Fig. 1: Comparison of the values of the moments of inertia determined  

by the effective and Harris method. 

The parameters of the model are determined in the following sequence: 

1. For the head energies ωK of the ground 
π

1
( 0 )K

+
=  and 

π
0

n
K

+
=  bands are using the 

experimental values 
exp.

( 0)
nK

E I =  because the head energies of these bands 0I =  (in 

this case rotω (0) 0= ) and these states are not mixed by the Coriolis forces; 

2. The head energies 
ν1

ω
 
of the 

π

ν
1K
+

=
 
bands found using their experimental values 

ν ν

эксп

1 1 rot
ω ( 1) ( 1);E I E I= = − =  

3. The head energies 
12

ω , 
22

ω of the 
π

2K
+

=
 
bands and 

1,2
ˆ2 1

x
j



+
   are free parameters, 

their values were obtained by the least squares method, from the condition of better 

agreement of the theoretical energy of states with odd spins 
π

1
2K
+

=
 
and 

π

2
2K
+

=  

rotational bands with the experimental data; 

4. The parameters ˆ0 1
n x

j


+
   describing the Coriolis interaction of the states of the 

π
0

n
K = , and 

π
1K


+
=

 
bands are determined from the good agreement of the calculated 

theoretical energies of the states with even spins of the ground
1

(0 ) , 
π

2 3 4 5
0 , 0 , 0 , 0K
+ + + +

=  

and 
π

1,2
2K
+

=
 
of bands with experimental values. 

The selected numerical values of the model parameters are given in Table 1. Figure 2 

shows the calculated energy values in the framework of the model which are compared with 

experimental results. 

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Fig. 2: Comparison of theoretical and experimental energy values. 

The results of the comparison show that at small values of spin, the compatibility 

between them is good, and the difference increases with increasing spin. This difference 

suggests that additional effects must be taken into account when studying the properties of 

high-spin states. For example, it may be necessary to take into account the effect of the 

rotational motion of the nucleus on the intrinsic energy when the rotation of the nucleus is 

high. 

The wave functions of the states 
π

2
0K
+

= , 
π

3
0K
+

= , 
π

4
0K
+

=  and 
π

1
2K
+

=
 
bands 

obtained by solving the Schrödinger equation (2) are present in tables 2, 3, 4 and 5 

respectively. The Hamiltonian basis includes 15 bands with 
π

1K


+
= , but the table shows 

one component of 
1

1K
 +
= . The components 

1

I

K


 
of the other 

π
1K


+
=  bands are 

determined using the following formula 

1

1

1

1 1

1

KI I

K K

K




 
 

 

−
=

− . (8) 

In Table 2, it can be seen that in the band with 
2

0K
 +
= , the other components 

20

I

K
  of 

the mixed bands increase with increasing spin I . This picture can be observed in other 
rotation bands. This result, which is due to Coriolis interaction, is the main cause of the 

nonadiabatics manifested in the energy and electromagnetic characteristics of the exited 

states. 

 

 

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Table 2: The amplitude of the mixing of the base states 
20

ψ
I

K
+  for 20K

 +
=  band 

K 

    I gr      02+    03+ 04+ 05+ 11+ 21+ 22+ 

2 -0.0052 0.9927 0.0261 0.0176 0.0068 0.0279 0.0927 0.0011 

4 -0.0183 0.9516 0.0808 0.0599 0.0231 0.0555 0.2534 0.0043 

6 -0.0385 0.8926 0.1404 0.1186 0.0460 0.0829 0.3496 0.0087 

8 -0.0638 0.8353 0.1872 0.1815 0.0708 0.1068 0.3890 0.0135 

10 -0.0922 0.7835 0.2177 0.2395 0.0942 0.1260 0.3988 0.0179 

12 -0.1222 0.7378 0.2352 0.2884 0.1143 0.1404 0.3951 0.0216 

14 -0.1529 0.6983 0.2440 0.3274 0.1306 0.1506 0.3859 0.0246 

16 -0.1836 0.6645 0.2476 0.3577 0.1436 0.1576 0.3750 0.0270 

18 -0.2141 0.6355 0.2482 0.3808 0.1537 0.1621 0.3639 0.0288 

20 -0.2444 0.6106 0.2469 0.3984 0.1615 0.1649 0.3533 0.0302 

 

Table 3: The amplitude of the mixing of the base states 
30

ψ
I

K
+  for 30K

 +
=  band 

K 

I gr      02+    03+ 04+ 05+ 11+ 21+ 22+ 

2 -0.0025 -0.0275 0.9986 0.0118 0.0044 0.0161 -0.0183 0.0007 

4 -0.0072 -0.0794 0.9923 0.0330 0.0122 0.0260 -0.0618 0.0023 

6 -0.0120 -0.1374 0.9795 0.0549 0.0204 0.0319 -0.1110 0.0040 

8 -0.0162 -0.1895 0.9625 0.0732 0.0272 0.0347 -0.1562 0.0054 

10 -0.0194 -0.2321 0.9446 0.0872 0.0324 0.0356 -0.1938 0.0064 

12 -0.0218 -0.2655 0.9278 0.0974 0.0362 0.0356 -0.2238 0.0072 

14 -0.0236 -0.2916 0.9129 0.1050 0.0391 0.0350 -0.2476 0.0078 

16 -0.0250 -0.3121 0.9001 0.1107 0.0412 0.0342 -0.2666 0.0082 

18 -0.0260 -0.3285 0.8890 0.1151 0.0429 0.0333 -0.2819 0.0086 

20 -0.0269 -0.3419 0.8795 0.1186 0.0442 0.0324 -0.2944 0.0088 

 

Table 4: The amplitude of the mixing of the base states 
40

ψ
I

K
+  for 40K

 +
=  band 

K 

I gr 02+ 03+ 04+ 05+ 11+ 21+ 22+ 

2 -0.0119 -0.0347 -0.0239 0.9486 0.1105 0.1673 -0.0186 0.0396 

4 -0.0302 -0.0952 -0.0675 0.8892 0.1910 0.2087 -0.0596 0.0546 

6 -0.0471 -0.1625 -0.1199 0.8468 0.2258 0.2115 -0.1055 0.0569 

8 -0.0604 -0.2265 -0.1744 0.8123 0.2393 0.2019 -0.1497 0.0566 

10 -0.0698 -0.2812 -0.2257 0.7809 0.2427 0.1886 -0.1881 0.0554 

12 -0.0760 -0.3249 -0.2708 0.7521 0.2414 0.1748 -0.2192 0.0539 

14 -0.0798 -0.3584 -0.3090 0.7262 0.2379 0.1620 -0.2436 0.0524 

16 -0.0822 -0.3838 -0.3408 0.7035 0.2337 0.1507 -0.2623 0.0510 

18 -0.0836 -0.4030 -0.3671 0.6839 0.2294 0.1409 -0.2767 0.0497 

20 -0.0844 -0.4177 -0.3890 0.6670 0.2254 0.1324 -0.2879 0.0486 

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Table 5: The amplitude of the mixing of the base states 
12

ψ
I

K
+  for 12K

 +
=  band 

K 

I gr      02+    03+ 04+ 05+ 11+ 21+ 22+ 

2 0.0021 0.0954 -0.0144 -0.0076 -0.0029 -0.0116 -0.9949 -0.0005 

3 - - - - - -0.0209 -0.9987 -0.0013 

4 0.0052 0.2667 -0.0352 -0.0191 -0.0073 -0.0168 -0.9620 -0.0014 

5 - - - - - -0.0328 -0.9967 -0.0032 

6 0.0070 0.3819 -0.0460 -0.0253 -0.0096 -0.0164 -0.9217 -0.0019 

7 - - - - - -0.0424 -0.9945 -0.0053 

8 0.0078 0.4465 -0.0510 -0.0283 -0.0108 -0.0149 -0.8920 -0.0021 

9 - - - - - -0.0504 -0.9921 -0.0076 

10 0.0082 0.4841 -0.0536 -0.0298 -0.0114 -0.0136 -0.8721 -0.0022 

11 - - - - - -0.0572 -0.9898 -0.0097 

12 0.0085 0.5078 -0.0551 -0.0307 -0.0117 -0.0124 -0.8585 -0.0023 

13 - - - - - -0.0630 -0.9875 -0.0119 

14 0.0087 0.5237 -0.0560 -0.0313 -0.0119 -0.0115 -0.8489 -0.0023 

15 - - - - - -0.0682 -0.9854 -0.0139 

16 0.0088 0.5350 -0.0566 -0.0317 -0.0121 -0.0108 -0.8418 -0.0024 

17 - - - - - -0.0727 -0.9833 -0.0158 

18 0.0089 0.5435 -0.0571 -0.0320 -0.0122 -0.0102 -0.8363 -0.0024 

19 - - - - - -0.0768 -0.9812 -0.0177 

20 0.0089 0.5500 -0.0574 -0.0322 -0.0123 -0.0097 -0.8321 -0.0024 

2 0.0021 0.0954 -0.0144 -0.0076 -0.0029 -0.0116 -0.9949 -0.0005 

 

4.   CONCLUSION 

Theoretical calculations were carried out for 
156

Gd  nucleus in the framework of the 

phenomenological model taking into account Coriolis mixing of low–lying rotation bands 

with positive parity. 

Nonadiabaticities observed in the energies is explained by the Coriolis mixing of low 

excited rotational states. To describe all the adiabatic rotational bands, the same moments 

of inertia have been used. The energy spectra of the positive parity states have been 

calculated. The results of calculation of energy spectra for ground π
1( 0 )K
+

= , 
π

20K
+

= , 

π
30K
+

= , 
π

40K
+

= , 
π

50K
+

= , 
π

12K
+

= , and 
π

22K
+

=  band states is compared with the 

existing experimental data correspondingly and their compatibility is given. The mixing 

effects of the lower bands have been shown to be significant even at small spins. 

In the high spin states of ground band, the difference between the theory and experiment 

was observed. This may be due to the fact that for large values of the angular frequency of 

rotation of the nucleus, it is necessary to take into account the effects of rotation on intrinsic 

energy. 

The wave function of states of the rotational bands is calculated. The regularities of the 

change in the state components of the mixing bands are studied depending on the total 

angular momentum.  

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ACKNOWLEDGEMENT  

This work has been financial supported by the Committee for the Coordination of the 

Development of the Science and Technology under the Cabinet of Ministers of the Republic 

of Uzbekistan, Fundamental Grant “OT-F2-75” (2017-2021). 

REFERENCES  

[1] Reich CW. (2012) Nuclear Data Sheets for A=156. 113: 2537-2840. 

[2] Backlin A, Hedin G, Folgelberg B, et al. (1982) Levels in 156Gd studied in the ( , γ)n  reaction. 

Nucl. Phys. A, 380: 189-260. 

[3] Berzin YY, Guseva TV, Tamberg YY. (1983) Investigation of 156Gd in the reaction ( , γ)n  

Izv. Akad. Nauk Lat. SSR, Ser. Fiz. Tekh. Nauk. 1: 9-12. 

[4] Bohle D, Richter A, Steffan W, et al. (1984) New magnetic dipole excitation mode studied in 
the heavy deformed nucleus 156Gd by inelastic electron scattering. Phys. Lett. B 137: 27-31.  

[5] Pitz HH, Berg UEP, Heil RD, Knaissl U, Stock R. et al. (1989) Systematic study of low-lying 
dipole excitations in 156,158,160Gd by photon scattering. Nucl. Phys. A, 492: 411-425. 

[6] Demidov А. М, Govor L. I, Kurkin V. А, Mikhailov I. V. (2009) Deexcitation of π
2 30 , 0 ,K
+ +

=

and 
12
+  levels by M1 transitions in nonspherical even–even nuclei. Phys. At. Nucl. 72(2): 201-

208.  

[7] Demidov АМ, Govor LI, Kurkin VА, Mikhailov IV. (2009) Confrontation of Nucleus 

Deformation and Multipole Mixture Parameters in 
1(2 0 2 0 )n

+ +
−  Transitions with Rotational 

Band Structure for π
20K
+

=  and 
π

30K
+

= . Phys. At. Nucl., 72(2): 209-213. 

[8] Aprahamian A, Lesher SR, Stratman А. (2017) First Excited 0+ States in Deformed Nuclei. 
Bulg. J. Phys. 44: 372--379.  

[9] Aprahamian A, de Haan RC, Lesher SR, et al. (2018) Lifetime measurements in 156Gd. Phys. 
Rev. C., 98:034303. 

[10] Usmanov P. N, Mikhailov I. N. (1997) Effects of nonadiabaticity of collective motion in even-
even deformed nuclei. Phys. Part. Nucl. 28(4): 348-373. 

[11] Usmanov PN, Okhunov AA, Salikhbaev US, Vdovin AI. (2010) Analysis of Electromagnetic 
Transitions in Nuclei 176,178Hf. Phys. Part. Nucl. Letters. 7(3):185–191. 

[12] Usmanov PN, Adam I, Salikhbaev US, Solnyshkin AA. (2010) Theoretical Analysis of 
Experimental Data on 160Dy That were Obtained in Studying Beta Decay. Phys. At. Nucl. 

73(12): 1990-1996. 

[13] Usmanov PN, Vdovin AI, Solnyshkin AA, Salikhbaev US. (2014) Properties of Оctupole - 
Vibrational Bands in the 160Dy Nucleus. Phys. At. Nucl. 77(11): 1343-1349. 

[14] Usmanov PN, Vdovin AI, Yusupov EK, Salikhbaev US. (2019) Phenomenological Analysis 
of Characteristics of Rotational Bands in 158,160Gd Isotopes. Phys. Part. Nucl. Letters. 19(6): 

706-712. 

[15] Harris SM. (1965) Higher Order Corrections to the Cranking Model Phys. Rev. B 138(3): 
509-513. 

[16] Bohr A, Mottelson B. (1977) Nuclear Structure, Vol. 1,2. New York: Benjamin 
[17] Bengtsson R, Frauendorf S. (1979) An interpretation of backbending in terms of the crossing 

of the ground state band with an aligned two–quasiparticle spectra band. Nucl. Phys., A 314: 

27-36. 

 

 

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