Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 51 (2018)

Eﬀect of pairing on the wobbling motion in odd-A nuclei†
K. Sugawara-Tanabe1,2 and K. Tanabe3
tion includes an important Coriolis anti-pairing (CAP)eﬀect,8) i.e., the Coriolis force originating from the rotation starts to dissolve the pair in the special highspin single-particle orbital, and ﬁnally the cranking
formula for MoI reduces to the rigid (rig) MoI. We
have obtained the analytical formula for the I dependence of MoI9) for both odd- and even-Z nuclei byapplying the second-order perturbation approximation
to the self-consistent Hartree-Fock-Bogoliubov (HFB)
equation under the number and I constraints. To
simulate the behavior of the I dependence of MoI,
we assume a two-parameter ﬁt for the rigid MoI J0 ,J0 (I − b)/(I + a) for highly excited states as in Lu
isotopes,6) and J0 /[1 + exp{−(I − b)/a}] for slightly
excited states as in 135 Pr.7)⃗ for the case
Figure 1 shows the alignments of R
135of the slightly excited states inPr, where the xaxis represents the maximum MoI. The parameter set
J0 =25 MeV−1 , a=7.5 and b=15.5 for j=11/2 simulates the experimental data quite well (see Figs. 17 and1/2
1/218 in Ref. 7)). Figure 1 shows that ⟨Rx2 ⟩I ∼ ⟨Rx2 ⟩I
1/2
1/2for I − j =even and ⟨Rx2 ⟩I+2 − ⟨Rx2 ⟩I ∼2. Therefore, the diﬀerence of ⟨Rx2 ⟩1/2 between the solid anddashed lines is almost one, indicating that the incremental alignment of ⟨Rx2 ⟩1/2 for I − j=odd is less byone unit compared with that for I − j=even, whichis associated with the excitation of the wobbling motion. A similar behavior is found for ⟨Ix2 ⟩1/2 in thisI-dependent rig MoI.
Because the wobbling mode is related to the rotational motion of the rotor, the RPA treatment, which
is useful for small-amplitude vibrational motion, is notapplicable to the wobbling mode.

As an indicator of a triaxial rotor, wobbling motion
was proposed by Bohr and Mottelson,1) and experimental data showing wobbling modes have been reported only in odd-Z nuclei of Lu isotopes,2) 167 Ta,3)
and 135 Pr.4) The wobbling motion is originally deﬁned
in classical mechanics5) as a precessional motion of angular momentum I⃗ around the axis either with themaximum or the minimum moment of inertia (MoI)of the rotating body. Quantum mechanically, the incremental alignment of I⃗ along the wobbling axis withthe maximum or the minimum MoI is in one unit1,6,7) .
In odd-Z nuclei, we found that in addition to the incremental alignment of I⃗ along the wobbling axis, the⃗ = I⃗ − ⃗j along the same
incremental alignment of R
axis is also in one unit (see Fig. 9 and Fig. 15 inRef. 7)), where ⃗j is the single-particle angular momentum. Moreover, the D2 invariance requires that theyrast wobbling band appears for the levels for which
I − j=odd.
The microscopic theory for nuclear rotational mo-Alignment

15Idependent Rig MoI 

R2x 1210   5
R2y 12         
2 12zReferences1) A. Bohr, B. R. Mottelson, in Nuclear Structure, (Benjamin, Reading, MA, 1975), II, Chap. 4, pp.190–194.
2) S. W. Ødegård et al., Phys. Rev. Lett. 86, 5866 (2001).
3) D. J. Hartley et al., Phys. Rev. C 80, 041304(R) (2009).
4) J. T. Matta, et al., Phys. Rev. Lett. 114, 082501 (2015).
5) L. D. Landau, E. M. Lifshitz, in Mechanics, (PergamonPress Ltd., Oxford/London, 1960), Sect. 37, pp.116–121.
6) K. Tanabe, K. Sugawara-Tanabe, Phys. Rev. C 77,064318 (2008).
7) K. Tanabe, K. Sugawara-Tanabe, Phys. Rev. C 95,064315 (2017).
8) B. R. Mottelson, J. G. Valatin, Phys. Rev. Lett. 5, 511(1960).
9) K. Tanabe, K. Sugawara-Tanabe, Phys. Rev. C 91,034328 (2015).

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101520IFig. 1. Alignments of ⟨Rx2 ⟩1/2 , ⟨Ry2 ⟩1/2 , and ⟨Rz2 ⟩1/2 for
the I-dependent MoI as functions of I. The solid andopen circles correspond to ⟨Rx2 ⟩1/2 and ⟨Ry2 ⟩1/2 , whilesolid and open triangles correspond to ⟨Rz2 ⟩1/2 . The
solid lines are for the levels with I − j=even, while thedashed lines for those with I − j=odd.
†∗1
∗2
∗3

Condensed from the talk in Int. Symp. on “Perspectives ofthe physics of nuclear structure”, Nov. 1-4, The Univ. ofTokyo (2017)
RIKEN Nishina Center
Department of Information Design, Otsuma Women’s University
Department of Physics, Saitama University- 96 -