Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 49 (2016)

Stability of the wobbling motion in an odd-A nucleus†
K. Tanabe∗1,∗2 and K. Sugawara-Tanabe∗1,∗3
Recently, the transverse wobbling mode was proposed in the yrast band near the ground state before
the ﬁrst backbending in 135 Pr1) . The transverse wobbling mode is the wobbling motion around the middle
moment of inertia (MoI)2) , which does not exist in thepure rotor as discussed in the context of classical mechanics3) and by Bohr-Mottelson4) quantum mechanically. Regarding the particle-rotor model, as long as
the rigid MoI is adopted, there is no chance to ﬁndtransverse wobbling, because the single-particle oscillator strength ωk and the rigid MoI are derived froma common radius, and their magnitudes increase or
decrease in the same direction as functions of γ periodically with a span of 2π/3. On the other hand,
the hydrodynamical (hyd) MoI changes its role in every span of π/3 in an opposite direction to ωk . Thus,
there remains a possibility to ﬁnd the transverse wobbling mode for the particle-rotor model with hyd MoI.
We extend the Holstein-Primakoﬀ (HP) boson expansion method to the odd-A case5–8) by introducing
two kinds of bosons for the total angular momentumI� and the single-particle angular momentum �j. We
can identify the nature of each band by referring totwo kinds of quantum numbers (nα , nβ ) which indicate
the wobbling of I� and the precession of �j, respectively.
In this paper we extend this method to the particlerotor model with hyd MoI. We choose a representation in which Iy and jx are diagonal because the hydMoI is maximum around the y-axis with the relation
Jyhyd ≥ Jxhyd ≥ Jzhyd in the range of 0 ≤ γ ≤ π/6,
while ωk favors �j to align along the x-axis in the samerange of γ. We notice that, if we choose the diagonal
representation for I� and �j in the same direction, wecannot ﬁnd any stable physical solution for common
values of γ and V (the strength of the single-particle
potential5–8) ) in the range of 11/2≤ I ≤33/2. We solve
the energy-eigenvalue equation to obtain two real solutions, i.e., ω+ , which is the higher energy, and ω− ,which is the smaller one. In the symmetric limit ofγ = 30◦ and V = 0, ω+ corresponds to the wobbling
motion around the y-axis with the maximum MoI,
while ω− corresponds to the precession of j aroundthe x-axis.
We adopt
j =11/2,
J0 =25 MeV−1 ( Jkhyd =
242γ + 3 πk ), β =0.18 and γ =26◦ (proposed
3 J0 sinby Ref.1) ), and V = 1.6 MeV (related to the singleparticle strength with these β, γ and j by Wigner†∗1∗2∗3E MeV10120exact
0,10,05

1015IFig. 1. Comparison of the excitation energy in the leading
order approximation5) with the exact results as functions of I. The solid lines correspond to I − j =even,the red dashed lines to I − j =odd, and the dotted bluelines to the exact results. The attached numerals (0,1)
and (0,0) correspond to quantum numbers of (nα , nβ ).

Eckart theorem)8) . In Fig. 1 we compare the energies
labeled by (nα , nβ )= (0,1) and (0,0) with the exact
ones obtained by diagonalizing the same Hamiltonian.
The exact levels are reproduced by the approximate
ones labeled (0,1), the precession mode of j, irrespective of I − j. Both ω+ and ω− monotonically increase
with I, and never decrease.
In conclusion, there is no transverse wobbling mode
within the framework of the particle-rotor model evenwith the hyd MoI.
References
1) J. T. Matta et al., Phys. Rev. Lett. 114, 082501 (2015).
2) S. Frauendorf, F. Dönau, Phys. Rev. C 89, 014322(2014).
3) L. D. Landau, E. M. Lifshitz: M echanics (PhysicsMathematics Institute, Moskow, 1958), Sect. 37.
4) A. Bohr, B. R. Mottelson: N uclear Structure (Benjamin, Reading, MA, 1975), Vol. II.
5) K. Tanabe, K. Sugawara-Tanabe, Phys. Rev. C 73,034305 (2006).
6) K. Tanabe, K. Sugawara-Tanabe, Phys. Rev. C 77,064318 (2008).
7) K. Sugawara-Tanabe, K. Tanabe, Phys. Rev. C 82,051303(R) (2010).
8) K. Sugawara-Tanabe, K. Tanabe, N. Yoshinaga, Prog.
Theor. Exp. Phys. 2014, 063D01 (2014).

Condensed from the article at the autumn meeting of Japan
Physical Society, Osaka City University, Sept. 27th (2015)
RIKEN Nishina Center
Department of Physics, Saitama University
Otsuma Women’s University- 81 -