June 2014, Tokyo

Advances in the microscopic
study of the Interacting
Boson ModelKosuke Nomura (GANIL)Interacting boson model (IBM)• Ingredients: collective pairsof valence nucleons• shell-model derivation. Validfor near vibrational nuclei
Microscopic basis
S pair(JP=0+)
D pair(JP=2+)

Nucleons boson(JP=0+)Refs:• A. Arima, F. Iachello (1974)• T. Otsuka, A. Arima, F. Iachello (1978)• T. Mizusaki, T. Otsuka (1996)•…

Otsuka-Arima-Iachello(OAI) mapping

full shell-modelconfig.

d boson(JP=2+)
BosonSD nucleon space

sd boson space

Q. Can we derive IBM Hamiltonian in some unified way ?

Long-standing problem“IBM may not be sufficient to describe some properties ofstrongly deformed nuclei because of the SD truncation.”
Many debates over the validity of the IBM. Concrete answer still
missing.
Nilsson-BCS model (T. Otsuka et al., 1982; D. Bes et al., 1982)• Renormalization of G pair or inclusion of g boson (e.g., T. Otsuka, Ginocchio,
1985; T. Otsuka, M. Sugita, 1988)• Conventional boson mapping (e.g., M. R. Zirnbauer, 1984)• J projection on the intrinsic wave function (N. Yoshinaga et al., 1984)• … Many others•

This talkA unified way of deriving IBM Hamiltonian for all
known cases of basic low-lying collective states:• Vibrational (weakly deformed) nuclei [1,2]• rotational (strongly deformed) nuclei [3]• Triaxial (or γ-soft) nuclei [4]• … (Other relevant cases)
Relevant publications:[1] K.N., N. Shimizu, T. Otsuka, PRL101, 142501 (2008)[2] K.N., N. Shimizu, T. Otsuka, PRC81, 044307 (2010)[3] K.N., T. Otsuka, N. Shimizu, L. Guo, PRC83 041302(R) (2011)[4] K.N., N. Shimizu, D. Vretenar, T. Nikšić, T. Otsuka, PRL108, 132501 (2012)

Principal idea

• A given mean-field model (Skyrme, Gogny, RMF, etc) can bea good starting point for computing spectra of the intrinsic
state of interest.

• We construct an IBM(-2) Hamiltonian by mapping themean-field solution onto the interacting-boson state.

?

Basic case: mapping energy surface

Mapping

Total energy of constrained
HFB within a given mean-fieldmodelTotal energy for a boson
condensation (energy expectation
value of boson system)

An IBM-2 Hamiltonian is determined by the equality EHFB≈EIBM; energy levels and wave functions with good spin, particle
number, parity, etc.

Fermion-boson mapping for deformed nuclei-

A basic rotational property should be also reproduced bybosons. We consider rotational response (= energy shift
due to infinitesimal rotation of nucleon intrinsic state).

-The missing piece is the rotational LL term in the IBM,giving correct moment of inertia.
- Intuitive picture -ricsnir
intdetotastateangle << 1intrinsic state at stationary

“Essential” boson Hamiltonian

Step 1) “Basic part”: mapping of the PES. Valid for nearspherical and γ-unstable cases

Step 2) LL term: rotational response (cranking) at a meanfield minimum. “Basic part” is kept unchanged. Necessary forstrongly deformed nuclei.

angle δθ << 1Proof of principle
Skyrme SkM*
134

BaExpt.

IBM

Mapped IBM
134

Ba
134

148

Sm154

SmBa

N

Spherical-Deformed transition in Sm;
Rotational band is reproduced nicely.
• Good description of γ-soft nucleus 134Ba•

r distinctions between these shapes. The data for even-even nuclei
ependence. It is found that nuclei with large asymmetries can be, a few percent deviation from y independence.

~r~-I.O

Pt

Xe 'V'Ba
Is a triaxial nucleus γ rigid-2.0 or unstable?
TRIAXIAL ROTOR"4+D-F: 1.67

≈ O(6)¥ - unstable( Wilets - Jean )y = 30 °50

S(Jγ,Jγ-1,Jγ-2)=[E(Jγ)-2E(Jγ-1)+E(Jγ-2)]/E(2+g)

1.08~j~y .

Lq5~- - 4O

. . . . .

5~m6;

6;

3o20.

.

.

.

3~

0°

10°

210°

- - - 0-ff~---- -202;

Y30°__4~ 3~

2~4~
Og

|_4

2

-1.0
-2.0

Xe-~'Ba '_W-J: -2.00 _,

I0A

-"""'
pli, ii,

i-30:50 50 70 90 110 130 150
Neutron Number

Fig. 1. Comparison of triaxial (or Davydov [6] ) rigid 7 rotor
softare halfway. Empirical models
and Majority
7-soft models. ofThe observed
Davydov resultsfor 7 =nuclei
3 0 ° are shown
explicitly
in themiddle forit.
comparisonwith the y-unstable,orcannot
explainMicroscopic
realization?
Wilets-Jean [ 7 ], model. (The y-band levels are shown as thickerlines. )Fig. 3. Empirical values of S(4, 3, 2 ) and S(6, 5, 4 ) f
even nuclei with Z> 30. Predictions of various model
the right, in particular the Davydov model (dash1958)• Rigid triaxial rotor model (Davydov & Filippov, on• γ-unstable rotor model (Wilets & Jean, 1956) scale ) and the 7-softmodel. For the latter, S is given aof 03/B (bottomscale for1980)
NB=6) by the solid liand O(6) inIBM (Ginocchio& Kirson,discussed
by Davydovbetween[6], and W-Jof a completelyy-flat• Equivalence(y-unstable) potential [ 7 ]. Since a nucleus having a

The “harmonic” structure of quasi-γ band isreproduced only if the IBM-2 Hamiltonian iscomprise of up to three-body boson terms.
2190+

Excitation energy (MeV)

8Skyrme SkM*+Os 8+6+5+8IBM-2 Mapped (2B)+

6+6+4+
3+5++

16+4++

433(4)

62108+4++

71.9(21)

0Expt.

5.9(6)+14.1101+14.4+105(6)2+64305++Skyrme SkM*

266.3108+

231022+27+

439102

8.2

4.1+

271.9

71.9

00++IBM(3B)

IBM(2B)

IBM-2 Mapped (2+3B)

Robustness• Independently of EDFs, neither W-J nor D-F picture isrealized in presumably all triaxial nuclei. New symmetry/phase?

• In the IBM, 3B term must be included.
134

S(4,3,2)21Ba(a)

196

Pt

194

132

Ba

Pt

192

192

Pt

Os

Jγ=4190

Os112

RuIBM(3B)

Three-body Ham.

0ï1IBM(2B)ï23WïJWïJ

2S(5,4,3)

DïF(a=//6)

IBM(2B)

10ï1
ï2

Two-body Ham.

Jγ=5(b)6IBM(3B)

91215N/Ni

1821

DïF
(a=//6)♦ Expt.
Skyrme (SkM*)
RMF (DD-PC1)

Shape coexistence in Hg
Mapped IBM
Gogny-D1M• IBM-2 with config. mixing (0p-0h +2p-2h) using Duval-Barrett’s technique• Spherical-Prolate-Oblate transition• Reasonable spectroscopic and intrinsic
properties

Application to “pear-shaped” nuclei

2+−

Ra−

13Excitation energy (MeV)

224

1211++

1012+12−11+101+−−8−7++5−
3−1+4+Expt.

76+4+2+0+8+−−

608+109+2+0−95−
3−
16+4+2+0Theo.

• Calc. by sdf-IBM mapped from RMF DD-PC1• Most pronounced octupole deformation at
226

Th, being a transition point.
• Good description of “octupole nucleus” 224Ra(cf. L. P. Gaffney et al. Nature (2013))

Summary
New formulation of the IBM has been developed. Bridge
the gap between IBM and nuclear mean-field model.
- Physical observables in lab frame- Valid for (in principle) any situation:• “Basic” part: mapping of energy surface. Static problem.
• LL term: mapping of rotational response. Dynamical problem.
• 3-B term: necessary for a triaxial minimum- Real prediction for heavy unknown nuclei- Remaining issues: mixed symmetry (in progress), superdeformation, … etc.

Thanks to
Main collaborators:
T. Otsuka (Tokyo)
N. Shimizu (Tokyo)
L. Guo (Tokyo, now in Beijing)
D. Vretenar (Zagreb)
L. M. Robledo (Madrid)
R. Rodríguez-Guzmán (Rice U)

European Commission forfinancial aidThank you for your attention!
ありがとうございます！