Vol. 40 (2009)ACTA PHYSICA POLONICA B

No 3

NEW FORMULATION OF INTERACTING BOSONMODEL AND HEAVY EXOTIC NUCLEI∗
K. Nomuraa , N. Shimizua , T. Otsukaa,b,c,d,ea

Department of Physics, University of Tokyo, Japan
Department of Physics, University of Tokyo, Japanc
Center for Nuclear Study, University of Tokyo, Japan
dRIKEN Nishina Center, Japane
National Superconducting Cyclotron Laboratory
Michigan State University, USAb

(Received October 28, 2008; revised version received December 13, 2008)
A novel way of determining the Hamiltonian of the interacting bosonmodel is proposed. The multi-fermion dynamics of surface deformations
studied by the mean-field theory, e.g., Skyrme model, can be mapped, ina good approximation, onto a boson system. The method is examined forwell-known nuclei, and predictions are presented for unexplored territoriesof the nuclear chart, namely, Os–W region nuclei with A & 200.
PACS numbers: 21.60.Ev, 21.10.Re, 21.60.Fw, 21.60.Jz

1. Introduction
The quadrupole collectivity is a prominent aspect in the nuclear structure for both stable and exotic nuclei and has been extensively studied interms of the interacting boson model (IBM) [1]. While the IBM has beensuccessful in describing the experimental data, the parameters of its Hamiltonian are in many cases determined phenomenologically. Besides that, theIBM has its own microscopic foundation, which has been studied by seniority
truncation using the shell model and applied only to spherical shapes [2–5].
For general cases, however, the microscopic foundation of the IBM remains
an open question. In this study, we propose a novel way of determining
the parameters of the IBM Hamiltonian for general cases, starting from themean-field model [6].
∗Presented at the Zakopane Conference on Nuclear Physics, September 1–7, 2008,
Zakopane, Poland.

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K. Nomura, N. Shimizu, T. Otsuka

2. Mean-field derivation of an IBM Hamiltonian
While the mean-field model, e.g., the Skyrme model, has been successful
in describing the intrinsic properties of all nuclei [7], one has not been able
to calculate in general the levels and wave functions of excited states withthe exact treatment of the angular momentum and the particle number [8].
Thereby, the Skyrme model seems to be insufficient for nuclear spectroscopy.
The IBM is a model of the quadrupole collectivity. Thus, it should be very
interesting to construct the IBM Hamiltonian based on the Skyrme model.
We first perform the constrained Skyrme Hartree–Fock+BCS calculations [9] and obtain the potential energy surfaces (PES’s). Once we get thePES (HF-PES, for brevity) in the mean-field calculations, we map a pointof it, denoted as (βBM , γBM ), onto a point of the boson PES (IBM-PES, forbrevity) in a good approximation.
In the present study, we discuss the IBM-2 [2], taking the standard
Hamiltonian Ĥ = ǫ(n̂dπ + n̂dν ) + κQ̂π · Q̂ν . The IBM-PES is calculated
as an expectation value of Ĥ in the boson coherent state [10]E(βb , γb ) = ǫ(nπ + nν )βb2 (1 + βb2 )−1 + 2nπ nν κβb2 (1 + βb2 )−2p×[2 − 2/7(χπ + χν )βb cos 3γb + χπ χν βb2 /7] ,(1)where βb and γb represent the intrinsic variables common to proton andneutron bosons. We also assume βb =Cβ βBM (with Cβ a coefficient) andγb =γBM for simplicity. The HF-PES is simulated by the IBM-PES using theparameters in Eq. (1), which are determined so that the latter reproduces
the former as well as possible. The overall pattern of the HF-PES reflects
the effects of nuclear force and Pauli principle for determining the energy ofcollective states. By reproducing the HF-PES, the IBM-PES is expected tosimulate, to a good extent, these effects in a simple manner.
(a)148Sm HF(SLy4) (b)148Sm IBM(g)134Ba HF(SkM*) (h)(c)‫ە‬‫ە‬154154Sm HF(SLy4) (d)134

Ba IBMSm IBM‫ە‬‫ە‬‫ە‬

(e)

132

132

Ba HF(SkM*) (f)‫ە‬BaIBM(i)208W HF(SkM*) (j)208W IBM[MeV]
2.01.5γΒΜγΒΜ

1.00.5

‫ە‬

0.0

0.1

0.2

0.3

βΒΜ

0.4

‫ە‬

0.0

0.1

0.2

0.3

βΒΜ (=βb/Cβ)

0.40.0‫ە‬

0.10.2

βΒΜ

0.0

‫ە‬

0.3 0.4 0.0 0.1

0.2

0.3

0.4

βΒΜ (=βb/Cβ)

Fig. 1. Comparisons of PES’s calculated by HF and IBM. Contour spacing is0.1MeV. Minima are identified by solid circles.

New Formulation of Interacting Boson Model and Heavy Exotic Nuclei

605

1.5
10.500−0.2(a)SmεκχνCβχπ =−0.5χν022+ +418892

Neutron Number

84

222+21+02+ 141+21+084

22+02+

1Cβ53(b)Sm(Expt.) (c)Sm(Th.) + (d)W(Th.)
02

20−13
SLy4

Energy[MeV]κ [MeV] ε [MeV]Fig. 1 depicts the comparisons of the PES’s, where the IBM-PES’s reproduce quite nicely the HF-PES’s for typical cases, i.e., near spherical(148 Sm), axially deformed (154 Sm) and γ-unstable (132,134 Ba and 208 W) nuclei. Fig. 2(a) exhibits the evolutions of the IBM parameters for Sm isotopes
with the neutron number N , reflecting the spherical-deformed shape transition. These variations of the parameters produce levels consistent withexperimental tendencies without adjustment to levels.

21+41+889296 848892

096 128 132 136 140Neutron Number Neutron Number Neutron NumberFig. 2. (a) Evolutions of the parameters in Eq. (1) with the neutron number N .
χπ is kept constant. (b) Experimental, (c) calculated (IBM from SLy4) levels forSm isotopes and (d) calculated (IBM from SkM*) levels for W isotopes with N .

3. Results
Figs. 2(b) and 2(c) show the low-lying levels of Sm isotopes as functionsof N , computed by the npbos code [11]. Around N = 86, the spectra
look like those of the U(5) limit in the IBM. As N increases, levels comedown consistently with the experimental trends particularly for yrast states.
Around N = 90, there seems to be the X(5) critical-point symmetry [12],+beyond which 0+
2 and 22 states go up in both experiment and calculation.
Finally around N = 92, the levels look like the rotational band (or SU(3)).
According to Figs. 1(e)–(h), 134 Ba has larger flat area than 132 Ba: 134 Ba
is more like the E(5) critical-point symmetry [13], while 132 Ba is closer toO(6). The calculated levels agree well with the experimental data.
Now we turn to the exotic W isotopes with A & 200. Figs. 1(i) and1(j) show HF- and IBM-PES’s for 208 W. Both PES’s have large flat areas
like 134 Ba, suggesting the E(5) symmetry. In Fig. 2(d), level evolution in W
(A & 200) isotopes is shown. Around N = 128, the levels look like those ofU(5) symmetry. As the magnitude of the deformation becomes larger withN , each level comes down, keeping the γ-unstable E(5)–O(6) pattern. Suchsustained E(5)–O(6) pattern has never been seen in stable nuclei and may
be a characteristic feature of exotic nuclei. Around N = 138, levels look likethose of γ-unstable nuclei, or O(6) limit.

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K. Nomura, N. Shimizu, T. Otsuka

4. Conclusions
In summary, we propose a novel way to derive the IBM Hamiltonian.
The mean-field model and the IBM can be complementary, where the former
plays a role to determine the parameters and the latter calculates levels andwave functions precisely. Using this method, low-lying collective states forthree dynamical symmetries, as well as recently proposed X(5) and E(5)critical-point symmetries. More importantly, we gain a capability to predictlevels and wave functions in unknown territories on the nuclear chart. Thiscan be a great advantage in the era of the third-generation rare-isotope
accelerators producing many new heavy exotic nuclei.
This work has been supported in part by the Mitsubishi Foundation, theJSPS core-to-core program (EFES) and the Inoue Foundation for Science.
REFERENCES[1] F. Iachello, A. Arima, The Interacting Boson Model, Cambridge UniversityPress, Cambridge 1987.
[2] T. Otsuka, A. Arima, F. Iachello, I. Talmi, Phys. Lett. 76B, 139 (1978);T. Otsuka, A. Arima, F. Iachello, Nucl. Phys. A309, 1 (1978).
[3] T. Mizusaki, T. Otsuka, Prog. Theor. Phys., Suppl. 125, 97 (1996).
[4] M. Deleze et al., Nucl. Phys. A551, 269 (1993).
[5] K. Allaart et al., Nucl. Phys. A458, 412 (1986).
[6] K. Nomura, N. Shimizu, T. Otsuka, Phys. Rev. Lett. 101, 142501 (2008).
[7] T.H.R. Skyrme, Nucl. Phys. 9, 615 (1959); D. Vautherin, D.M. Brink, Phys.
Rev. C5, 626 (1972).
[8] J. Dobaczewski et al., Phys. Rev. C76, 054315 (2007); T. Duguet et al., private
communication.
[9] P. Bonche et al., Comput. Phys. Commun. 171, 49 (2005).
[10] A.E.L. Dieperink, O. Scholten, Nucl. Phys. 346, 125 (1980); J.N. Ginocchio,M. Kirson, Nucl. Phys. A350, 31 (1980); A. Bohr, B.R. Mottelson, Phys. Scr.
22, 468 (1980).
[11] T. Otsuka, N. Yoshida, JAERI-M report 85 (1985).
[12] F. Iachello, Phys. Rev. Lett. 87, 052501 (2001).
[13] F. Iachello, Phys. Rev. Lett. 85, 3580 (2000).