Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 48 (2015)

Nuclear moment of inertia in super-normal phase transition region†
K. Tanabe∗1,∗2 and K. Sugawara-Tanabe∗1,∗3

†
∗1
∗2
∗3

Condensed from the article in Phys. Rev. C, Vol. 91, 034328(2015)RIKEN Nishina Center
Department of Physics, Saitama University
Otsuma Women’s University

1.04

odd mass

1.02

JJrig

The purpose of this paper is to derive the analytic
expression for the angular momentum (I) dependenceof the moment of inertia (MoI) from the microscopic
many-body theory both for even-even and odd-massnuclei. The I-dependence of MoI has been proved tobe essential in simulating triaxial, strongly deformed(TSD) bands in a series of papers.1–4)
We adapt the approximation developed for the gap(Δ) dependence of the ratio of MoI (J) to the rigidbody value (Jrig ).5,6) It assumes that only large matrix elements of single-particle angular momentum of(jx )αβ contribute to J with a common excitation energy of δ(= εβ − εα ), where εα denotes the singleparticle energy of the level α. We apply this approximation to the gap equation including the Coriolis antipairing (CAP) eﬀect7) through the second-order perturbation to the cranking term.8,9)
When Δ is larger than half of the single-particle level
distance d, we can apply a deﬁnite integral for the gap
equation with the CAP eﬀect. When Δ is smaller thanhalf of d, we propose the ﬁnite sum method with thepicket-fence approximation for the level distribution.
In this case, it is proved that Δ never tends to zero,and there is no sharp phase transition from the superconducting state to the normal state. Neglecting thehigher order in 2Δ/δ for the case Δ < d/2 (ﬁnite summethod), we express MoI as an analytic function of I.
In Fig. 1, we compare the approximate solution between even-even and odd-mass nuclei as functions of I
measured from the band-head angular momentum I0 .
Usually, I0 = 0 for even-even nucleus, while I0 �= 0for odd-mass nucleus, for example, I0 = 13/2 for theTSD yrast band in 163 Lu.10) We choose the singleparticle energy for a valence nucleon as ε� = 0.6 MeV
above the Fermi surface, and the initial pairing gap at
I=I0 for odd mass as 0.6 MeV, smaller than 0.8 MeVfor even-even nucleus (blocking eﬀect). The blockingeﬀect reduces the starting value of Δ and increases
that of the MoI. In odd-mass case, there is a termthat correlates the single-particle state of � with αthrough (jx )2α� . The matrix element of (jx )2α� is chosen to be 12 for εα > ε� and 10 for εα < ε� . The
other parameters are the same as those for the eveneven case. We have started both approximate solutions
with Δ = 0.15 MeV corresponding to I−I0 ∼15, whiled = 0.4 MeV.
As is seen in Fig. 1, the main diﬀerence between
even-even (dashed line) and odd-mass (solid line) nu-1

eveneven

0.98
0.96
0.94
0.92
15

20

25

30

35

40

II0Fig. 1. Comparison of the ratio J/Jrig in the approximate
sum method as functions of I-I0 for even-even (dashedline) and odd-mass (solid line) nuclei.

clei is from the blocking eﬀect. Then, both curves increase gradually, and approach the value 1. The MoIof odd-mass case is chosen to be slightly larger thanthat of the even-even case. The curves become convex upward before they reach to rigid-body values.
This upward convexity is also necessary for explaining the energy sequence of TSD bands.4) For the caseof Δ ≥ d/2 (deﬁnite integral), J goes to Jrig around
I−I0 ∼ 17 or 18 (sharp phase transition). Even in thiscase, odd-mass nuclei show an upward convexity before the phase transition at I=17 ∼ 18. Because oflarger I0 , the slow phase transition occurs at larger Ifor odd-mass nuclei than for even-even nuclei.
References
1) K. Tanabe and K. Sugawara-Tanabe: Phys. Rev. C 73,034305 (2006).
2) K. Tanabe and K. Sugawara-Tanabe: Phys. Rev. C 77,064318 (2008).
3) K. Sugawara-Tanabe and K. Tanabe: Phys. Rev. C 82,051303(R) (2010).
4) K. Sugawara-Tanabe, K. Tanabe and N. Yoshinaga:
Prog. Theor. Exp. Phys. 2014, 063D01 (2014).
5) A. Bohr and B. R. Mottelson: N uclear Structure (Benjamin, Reading, MA, 1975), Vol. II.
6) D. Bengtsson and J. Helgessen: Lecture notes in a summer school at Oak Ridge (1991).
7) B. R. Mottelson and J. G. Valatin: Phys. Rev. Lett. 5,511 (1960).
8) M. Sano and M. Wakai: Nucl. Phys. 67, 481 (1965).
9) K. Sugawara: Prog. Theor. Phys. 35, 44 (1966).
10) S. W. Ødegård et al .: Phys. Rev. Lett. 86, 5866(2001).

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