M Singh

Rajesh Kumar

S Sharma

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Study of odd-even staggering in γ-band of 154Gd 
 
Rajesh Kumar1*, M. Singh1 and S. Sharma2** 
1Department of Physics, Noida Institute of Engineering & Technology, Greater Noida-201306, INDIA 
2Department of Physics, Yobe State University, Damaturu, NIGERIA 
*Email: rajeshkr573@gmail.com    **Email: satendrasharma@ysu.edu.ng 
 
Introduction 
The 154Gd nucleus (Z=64, N=90) is lying 
at the transitional position between the 
deformed and spherical nuclei. The nuclear 
shape for N≤88 nuclei is close to spherical and 
for  N≥90 nuclei it is close to deformed [1, 2]. 
For 154Gd nucleus, the observed energy ratio R4 
is equal to 3.01 which is close to X(5) 
symmetry limiting value of R4 equal to 2.9 [3]. 
The existence of X(5) symmetry in the N=90 
isotones (Sm-Gd-Dy) established in recent 
work [4] also supported by the formation of 
isotonic multiplets [5]. The value of R4 in the 
present IBM-1 calculation is 3.2 which is 
slightly more than the experimental value. The 
theoretical studies of level energies of γ-band 
(Kπ=2+) of shape transitional nuclei are limited. 
The idea of odd-even spin staggering (OES) in 
γ-band was given by McCutchan et al. [6]. The 
structure of γ- vibrational and quasi γ- bands of 
even- even nuclei was investigated by Gupta et 
al. [7] on a global scale. The yrast and γ -bands 
of 120−130Xe nuclei was studied by Singh et al. 
[8] using asymmetric rotor model by 
employing the Lipas parameter  and pointed 
out that the reason of odd even staggering 
(OES) was due to  the splitting of γ -band in 
odd and even spin sequence. Recently, Gupta et 
al. [9] illustrated that the values of odd-even 
energy staggering index S(4) is small for well 
deformed nuclei and the odd-even spin 
members form a single K=2 band. 
 
In the present work, we test the odd-even 
spin staggering (OES) in γ-band in 154Gd 
nucleus in the framework of rigid triaxial rotor 
model (RTRM) [10] and interacting boson 
model (IBM-1) [1]. In the present empirical 
work, we also studied that whether this nucleus 
is γ-soft or γ-rigid? 
 
 
Odd-even staggering (OES) 
The OES effect represents the relative 
displacement of the odd angular momentum 
levels of the γ- band with respect to their 
neighboring levels with even angular 
momentum. The band mixing interaction 
pushes the even spin members in γ-band 
relative to the odd spin members, due to the 
interaction with even spin members of the 
ground band [1]. The staggering in band 
energies and the transition between different 
structural symmetries in nuclei is calculated by 
using the expression [4]: 
 
[ ] [ ]
+
------
=
12
)2()1()1()(
)(
E
JEJEJEJE
JS  (1) 
Therefore, using equation (1) we have 
calculated the value of staggering index S(J) 
for 154Gd. The index of odd-even spin 
staggering is a quantitative measurement of 
OES with spin. 
 
Result and discussion 
The experimental energies of γ-band for 
154Gd have been taken from the Ref. [11]. The 
values of energies of γ-band are calculated 
from IBM-I and RTRM. The IBM-1 
Hamiltonian (with OCT=HEX=0) is used to 
reproduce the best energy spectrum (see Ref. 
[12] for details). The fitting parameters (in 
MeV) are EPS= 0.3425, PAIR= 0.0116, ELL= 
0.0128 and QQ=-0.0221. The energies 
calculated in RTRM [10] are normalized using 
Lipas like relation: 
RTRM
RTRM
E
E
fitE
.1
)(
a+
= . 
Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 60 (2015) 166
Available online at www.sympnp.org/proceedings
Where, the Lipas parameter ( )a  used for 
normalization of energies is 4.5x10-5 for γ-
band. The S(J) for γ–band is calculated using 
equation (1) and listed in Table 1. The IBM 
index of odd-even spin staggering S(J) with 
spin (J) are much larger with corresponding 
experimental and RTRM values. 
The plot of experimental index of odd-even 
spin staggering S(J) with spin (J) tallies with 
that of our calculated  RTRM values (see 
Fig.1). The all S(J) values of RTRM are 
positive and a small in magnitude except S(7) 
and show slightly zig-zag behavior, but 
increase slowly with increase in spin. This 
indicates that 154Gd nucleus is axial symmetric 
rotor. The staggering index S(4) is 0.33 for 
axially symmetric rotor and 1.67 for rigid 
triaxial rotor. For γ -soft rotor or O(6), S(4) = 
−2.0 and for spherical vibrator it is = −1.0 [8].  
In our RTRM calculations the value of S(4) is 
0.34 (see Table 1) which tally with axially 
symmetric rotor value of 0.33. This further 
supports the axial symmetric deformed rotor 
nature of 154Gd nucleus.  
 
Table 1. The values of odd- even spin 
staggering in γ-band for 154Gd as estimated in 
the RTRM and IBM-1 along with experimental 
results. 
 
S(J)à S(4) S(5) S(6) S(7) S(8) 
EXP. 0.03 0.26 0.04 0.25 - 
RTRM 0.34 0.17 0.50 -0.17 0.95 
IBM-1 1.96 -1.78 2.75 -2.65 3.62 
 
Acknowledgement 
 
We are grateful to Prof. J. B. Gupta, Ramjas 
College, University of Delhi, for fruitful 
discussion. RK is wishing to express his 
gratitude to Dr. O. P. Agrawal, Chairman, and 
Dr. Ajay Kumar, Director, NIET, Gr. Noida for 
providing the research facilities. SS is grateful 
to Prof. Musa Alabe, Vice Chancellor, Yobe 
State University, Damaturu, Nigeria for 
encouragement and for providing the research 
facilities. 
 
-4
-2
0
2
4
154Gd
S(8)S(7)S(5) S(6)S(4)
 Exp
 IBM
 RTRM(fit)
S
ig
na
tu
re
 s
pl
iti
ng
Spin (J)
 
Fig.1 The variation of S(J) versus spin (J). 
 
References 
 
[1] R F Casten, Nuclear structure from a simple 
perspective (Oxford University Press, New 
York, 1990). 
[2] Rajesh Kumar, S Sharma and J B 
Gupta, Arm. J. Phys.  5 (2), pp. 62-74, 2012. 
[3] F Iachello, Phys. Rev. Lett. 87, 05-2502, 
2001. 
[4] J B Gupta, Eur. Phys. J. A. 48, 177, 2012. 
[5] Rajesh Kumar and S Sharma, Can. J. Phys. 
93, 711-715, 2015.  
[6] E A McCutchan, et al., Phys. Rev. C 76, 
024306, 2007. 
[7] J B Gupta and A K Kavathekar, Pramana J. 
Phys. 61, 167, 2003. 
[8] M Singh et al. Can. J. Phys. 85, 889, 2007. 
[9] J B Gupta, S Sharma and V Katoch, 
Pramana J. Phys. 81, 75, 2013. 
[10] A. S. Davydov and G. F. Filippov, Nucl. 
Phys. 8, 237, 1958. 
[11] http://www.nndc.bnl.gov/nsdf (2015) 
[12] Rajesh Kumar, S Sharma and J B Gupta, 
Advances in Applied Science Research, 3 (3), 
1744-1748, 2012.   
Proceedings of the DAE-BRNS Symp. on Nucl. Phys. 60 (2015) 167
Available online at www.sympnp.org/proceedings


Study of odd-even staggering in γ-band of 154 Gd

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Study of odd-even staggering in γ-band of 154 Gd

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