Microscopic description of shape coexistence/mixing phenomena
as large-amplitude collective motions
Koichi Sato (Theoretical Nuclear Physics Lab., RIKEN / Department of Physics, Kyoto Univ.)
OutlineAtomic nuclei exhibit different shapes depending on the numbers of protons and neutrons, the excitation energies, or angular momentum.

Collaborators:

Shape coexistence phenomena, in which an excited band with a shape different from the shape in the ground band exists close to the groundband, are widely observed all over the nuclear chart. Since the nuclear deformation is in general sensitive to the nuclear structure, theN. Hinohara(RIKEN), M. Matsuo(Niigata Univ.),

description of these shape coexistence phenomena can be a touchstone for nuclear structure models. In this study, we mainly study theT. Nakatsukasa (RIKEN),K. Matsuyanagi(RIKEN/ YITP,Kyoto Univ.)oblate-prolate shape coexistence/mixing as large-amplitude collective motions in the low-lying states of proton-rich Se-Kr isotopes on thebasis of the adiabatic self-consistent collective coordinate (ASCC) method.

Shape coexistence phenomena

Microscopic approach to large-amplitude collective motions

Atomic nuclei exhibit different shapes depending on the numbers of protons and neutrons, the excitation

The construction of a microscopic theory of large-amplitude collective motions is a long-standing
open problem in nuclear structure physics.
In this work, we attempt to construct a microscopic theory of large-amplitude collective motions
on the basis of the adiabatic self-consistent collective coordinate(ASCC) method.

energies, or angular momentum. Shape coexistence phenomena, in which an excited band with a shapedifferent from the shape in the ground band exists close to the ground band, are widely observed all overthe nuclear chart.

e.g. shape coexistence in 186Pbhpatev
ctiecoll

The ASCC method is based on the time-dependent
Hartree-Fock-Bogoliubov (TDHFB) method andan adiabatic solution of the SCC method formulated

ingppmacollective hypersurface

by Marumori et al, with which we can extract theoptimal collective path (collective submanifold),maximally decoupled from non-collective degrees ofhuge-dimensional
TDHFB conﬁguration space

freedom, in the large-dimensional TDHFB phasespace.
T. Marumori et al., Prog. Theor. Phys. 64 (1980), 1294

A. N. Andreyev et al., Nature 405, (2000) 430-433.

M. Matsuo et al., Prog. Theor. Phys. 103 (2000) 959.

shape coexistence and transition in proton-rich Kr isotopes
Appearance of a low-lying excited 0+ state is one of
typical experimental indications of the shape

shape coexistence/ mixing in proton-rich Se and Kr isotopes and gave successful results. Now we are going toextend the 1D ASCC method to the two-dimensional version.

Low-lying excited 0+ states
Irregularities of the excitation spectra
Systematics of the excited 0+ energies and E0 transition strengths

0.085(19)0.079(11)Adiabatic Self-consistent Collective Coordinate method
The one-dimensional (1D) ASCC method, in which we assume one collective coordinate, was applied to thecoexistence.

0.072(6)N. Hinohara et al., Prog. Theor. Phys. 117, 451 (2007).

collective potential energy map
in the β − γ plane

N. Hinohara et al., Prog. Theor. Phys. 119, 59 (2008); Phys. Rev. C 80, 014305 (2009).

As a first step toward the full 2D ASCC method, in this study, we solve the two-dimensional ASCC approximately
and determine microscopically the three vibrational and three rotational inertial functions and the collective
potential in the five-dimensional quadrupole collective Hamiltonian: we solve the constrained

0.047(13)Hartree-Fock-Bogoliubov (CHFB) equation and the “local” QRPA (LQRPA) equations at each point on the β − γ plane.

C. Chandler et al. PRC 56 (1997) 2924.
E. Bouchex et al., PRL 90 (2003) 082502.

Constrained HFB + Local QRPA method

Tunneling effects and shape mixing

5D Quadrupole Collective Hamiltonian(General Bohr-Mottelson Hamiltonian)

From the mean-field viewpoint, shape coexistenceSkyrme-HFB

indicates that there are two equilibrium points in the

A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952).
A. Bohr and B. R. Mottelson, Mat. Fys. Medd. Dan. Vid. Selsk. 27,No.16(1953).

mean-field potential.

H = Tvib + Trot + V ( β , γ )The nucleus is a quantum many-body system:
Different shapes are mixed by the tunneling effectvibrational energy:

112Tvib = Dββ ( β , γ ) β + Dβγ ( β , γ ) βγ + Dγγ ( β , γ )γ 222

through the potential barrier between the minima.

Shape mixing by tunnelingbetween the local minima

Collective potential

many-body tunnelingrotational energy:

vibrational inertial mass

1
2
Trot = ∑ J k ωkk =1 2
3

Large-amplitude collective motion

Local QRPA(LQRPA) mass

Moment of inertia

quantization
Collective Schrödinger equation

rotational energy[Tˆ (β , γ ) + Tˆ (β , γ ) + V (β , γ )]Ψ (β , γ , Ω) = E ΨI ( I + 1)
E(I ) =2Jvib

IMαrotEM transition caused by shape mixing

Local QRPA masses
rotational

vibrational

IM αΨIMα ( β , γ , Ω) = ∑ Φ IKα ( β , γ ) Ω IMK
K :evenoblate
γ = 60

KS & N. Hinohara, arXiv:1006.3694

vibrational

γrotationaltaken from L. Fortunato, Eur. Phys. J. A 26 (2005) 1.

β − γ plane
Strong dependence on (β , γ ):the 2nd lowest minimum

Excitation Spectra & E2-transition
properties
Excitation Energy [MeV]

4Inglis-Belyaev

3.5415362.522748
4+1412225+
21301

2732+

221042+970 02+

0177

4260

2+
1+

01812243 668

61+2412271901

21660
1229++
245+41+1274

localization

+2416
66

3201

61

21

0.5
0

4
640

1.5
1139

32112+

02

spherical

γ = 0

βprolate

Summary
We have developed a new method to determine microscopically the 5D
quadrupole Hamiltonian on the basis of ASCC method for the description oflarge-amplitude collective motions in atomic nuclei
Numerical results for proton-rich Kr isotopes show that the rotationalmotion plays an important role for the growth of the deformation.

Outlook+

623353+
1

Collective wave functions squaredβ(probability densities)

Experiment

LQRPA+

62

The residual interaction
increases the inertial masses

4EXP：Fischer et al., Phys.Rev.C67 (2003) 064318,
Bouchez, et al., Phys.Rev.Lett.90 (2003) 082502.
Gade, et al., Phys.Rev.Lett.95 (2005) 022502, 96 (2006) 189901

triaxial

Effects of residual interaction

β =0:the absolute minimum( β , γ , Ω)deviation from the regular rotational spectra

Application of the CHFB+LQRPA method to 72Kr
Collective potential

I ,αΩ = (θ1 , θ 2 , θ3 ) : Euler angles

Collective wave function
S.M. Fischer et al., Phys. Rev. Lett. 84, (2000) 4064.

One of the most important features of ourmethod is that the inertial masses obtained with
our method contain the contribution from theresidual interaction unlike the widely-usedcranking masses.

Fully self-consistent calculation:
We are going to extend the CHFB+LQRPA method, an approximate version of
the two-dimensional ASCC method, to the fully self-consistent version.

++21+021000(13)+01The numbers on arrows show B(E2) values in units of e fm .
24Rotation hinders shape mixing

Realistic interaction:
We are going to adapt a more realistic interaction like Skyrme interaction, which
is one of the standard interactions in modern nuclear structure physics , based on
the density functional theory