Bulg. J. Phys. 42 (2015) 344–353

Microscopic and Algebraic Theory forComplex Nuclear Shapes and Dynamics
K. Nomura1,2,3
1Grand Accélérateur National d’Ions Lourds, CEA/DSM-CNRS/IN2P3,F-14076 Caen Cedex, France
2
Institut de Physique Nucléaire, CNRS/IN2P3, F-91406 Orsay Cedex, France
3
Physics Department, Faculty of Science, University of Zagreb,
HR-10000 Zagreb, Croatia
Received 10 October 2015
Abstract. An overview is given on the advances in the recently developed
method of merging nuclear energy density functional theory and the interactingboson model. By establishing the mapping between the fermionic and bosonic
intrinsic states, the Hamiltonian of the interacting boson model is derived forgeneral cases covering from vibrational to rotational deformed regimes. Recent examples resulting from the method are reported: the phenomena of shapecoexistence and octupole deformation.
PACS codes: 21.60.-n1Introduction

The atomic nucleus, although it is a highly quantum mechanical system consisting of protons and neutrons, is a bound object demonstrating stunning collective excitation associated with certain geometrical shape [1]. The nuclear
shape arises due to interactions among nucleons and quantum many-body effects (mean field, coherence, superconductivity etc). The shape can change withthe addition/subtraction of nucleons and result in a beautiful regularity in its(vibrational and rotational) energy spectrum despite the underlying complex nuclear correlations. Such self-consistent shape can be actually observed in various
quantum many-body systems, including nuclei, molecules and condensed matter, and common theoretical methods are used. Thus, the description of nuclear
shapes and collective excitations only from nucleonic degrees of freedom has
been one of the central issues of nuclear theory.
The interacting boson model (IBM) of Arima and Iachello [4] is well established, based on group theory, for a complete and straightforward description ofnuclear low-energy excitations. The IBM is a model where BCS-like pairs of nucleons with spin J = 0+ (S pair), 2+ (D pair), . . . are approximated by bosons344

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Microscopic and Algebraic Theory for Complex Nuclear Shapes and Dynamics
with J = 0+ (s boson), 2+ (d boson), . . .. It in fact presents a vast truncationof (shell-model) configuration space, while keeping the essence of low-energy
nuclear structure. The IBM itself has been, however, a phenomenological modelso that the parameters of its Hamiltonian have to be taken from experiment. For
this reason, many attempts have been made to establish a link between the IBM
and a more microscopic nuclear structure model. The prescription widely taken
is the one developed by Otsuka, Arima and Iachello [3], known as the OAI mapping, where a seniority-based shell-model state, expressed by low-spin S, D, . . .
pairs, is mapped onto the equivalent s, d, . . . boson states. This method is valid
for but is limited to near-spherical and γ-soft shapes, because the underlying
shell-model configuration becomes highly complicated for deformed cases. The
microscopic justification of the IBM for general shapes, particularly strongly deformed ones, was questioned [4] and has been under an active investigation up
to the 80’s, but since then without much progress.
An entirely new method of deriving the IBM has been presented in [5], wherethe authors proposed to derive the IBM Hamiltonian by mapping the potential energy surface (PES), calculated within the self-consistent mean-field approach on the basis of a microscopic non-relativistic or relativistic energy density functional (EDF), onto the bosonic counterpart. The EDF framework, witha given parametrization, provides an accurate and global description of the intrinsic properties of nuclei [6], and the IBM has been shown to be derived fromthere in principle for any shapes.
In this contribution, we report recent advances in this approach, demonstrating
first how it works in basic quadrupole collective modes – vibrational, rotationaland γ-soft regimes as the proof of principle and then applications to shape coexistence and octupole collective states.
2Theoretical Framework

2.1

Mean-field calculation

The first step is a standard constrained mean-field calculation, e.g., in the
Hartree-Fock-Bogoliubov (HFB) framework, based on a given relativistic or
non-relativistic EDF that is successful in a global description of nuclear structure
phenomena. The constraints are on mass quadrupole moments related to the axial deformation β and non-axial deformation γ [1]. For a given set of collective
coordinates (β, γ), the HFB calculations are performed to obtain PES, an example of which is shown on the left-hand side of Figure 1 for axially-deformed
148,150,154
Sm and γ-soft 134 Ba nuclei. In these particular cases, the calculation has been made with the constrained Hartree-Fock plus BCS (HFBCS)
method [7] using the Skyrme [8, 9] EDF with the SkM* parametrization [10].
One sees in the figure that the HFBCS calculation predicts the intrinsic shapesof the 148,150,154 Sm nuclei to be the representatives for the near-spherical vibra345

K. Nomura

Figure 1. (Color online) βγ PESs of 148,150,154 Sm and 134 Ba isotopes. On the left-handside are microscopic HFBCS PES with the Skyrme SkM* interaction, while on the righthand side are the mapped IBM PESs. The contour lines join points with the same energy(in MeV) and the color scale varies in steps of 50 keV. The energy difference between
neighboring contours is 0.5 MeV. The minimum is identified by the filled circle.

tional, transitional and rotational deformed nuclei, respectively. One sees a nearspherical minimum for 148 Sm, finds a deep minimum at β ≈ 0.35 for 154 Sm,and for 150 Sm a minimum much softer in β direction. For the 134 Ba nucleus, theHFBCS calculation provides the PES almost totally flat in γ deformation with ashallow triaxial minimum.
2.2

Mapping onto the boson space

Having the HFB PES hφf (β, γ)|Ĥf |φf (β, γ)i for each individual nucleus, with|φf (β, γ)i representing the HFB solution for a given deformation (β, γ), we346

Microscopic and Algebraic Theory for Complex Nuclear Shapes and Dynamics
then equate it to the expectation value of a given IBM Hamiltonian in the boson intrinsic state. The boson intrinsic state, denoted as |φb i, isPwritten as[11] |φb i = (Nπ !Nν !)−1/2 (λπ )Nπ (λν )Nν |oi, where λ†τ = s†τ + µ ατ µ d†τ µ .
The coefficients ατ µ are given as ατ ±2 = 2−1/2 βτ sin γτ , ατ ±1 = 0 andατ 0 = βτ cos γτ . |φb i invokes deformation variables for both protons (βπ , γπ )and neutrons (βν , γν ). Since in the medium-heavy and heavy nuclei, protons andneutrons attract each other strongly, the proton and neutron systems should havethe same intrinsic shape in first approximation, leading to βπ = βν ≡ βb andγπ = γν ≡ γb . In addition, the variable βb in the boson system can be associatedwith the β deformation in the collective model so that they are proportional toeach other [11], β ∝ βb , while γb can be taken to be the same as the γ parameterin the collective model as both have the same meaning.
Under these conditions, and within a relevant range of (β, γ), i.e., around theglobal minimum, we equate the HFB and the IBM PESs
hφf (β, γ)|Ĥf |φf (β, γ)i ∼ hφb (βb , γb )|Ĥb |φb (βb , γb )iwith

β = βb /Cβ ,andγ = γb ,(1)(2)

where Cβ is a coefficient. The parameters of the IBM Hamiltonian are determined through this procedure so that the topology of the HFB PES in the vicinity of the energy minimum is reproduced by the IBM. We restrict ourselves tothe region only close to the energy minimum, because it is most relevant to thelow-energy collective states, while one should not try to reproduce the region far
from the energy minimum as it is dominated by single-particle configurations
that are outside the model space of the IBM by construction. The topology of
the HFB PES should reflect essential features of fermion many-body systems,
such as the Pauli principle, the antisymmetrization, and the underlying nuclear
force etc. Through the mapping procedure, these effects are supposed to besimulated by the IBM system.
The IBM Hamiltonian which embodies the essentials of the underlying
fermionic interactions is taken to be of the form [3, 4]
Ĥb = n̂d + κQ̂π · Q̂ν ,(3)where the first and second term represents, respectively, the d-boson numberoperator (n̂d = n̂dπ + n̂dν ), with  being the single d-boson energy relative tothe s-boson one, and the quadrupole-quadrupole interaction with strength κ. Thequadrupole operator is Q̂τ = d†τ sτ + s†τ d˜τ + χτ (d†τ d˜τ )(2) (τ = π or ν) with χτbeing a parameter. The model parameters, , κ, χπ , and χν plus the coefficientCβ (see Eq. (2)), are fitted to the HFB PES (for details, see [12]). On the righthand side of Figure 1, we observe that the mapped IBM energy surfaces of all thenuclei considered reproduce the topology of the HFB (or HFBCS) counterpartsin the vicinity of the absolute minima. Compared to the microscopic PES, the347

K. Nomura
mapped IBM one looks generally flat in the region far from the minimum. Thisis a consequence of the limited model space and/or the finite boson number in
the IBM framework.
When a nucleus is strongly deformed, the way the fermion system responds torotation has been shown to be significantly different from the boson system [13],and this difference results in the systematic underestimation of the rotational
moment of inertia. The reason is the limited boson content (only s and d bosons)in the IBM, while the PES is correctly mapped with s and d bosons only. In orderthat the rotational responses of fermion and boson systems agree with each other,
the term L̂·L̂ was introduced in the IBM [13], and this term should be included in
the Hamiltonian in Eq. (3) for a complete description of quadrupole modes. TheL̂ · L̂ term modifies the moment of inertia without disturbing other parameters
already fixed through the PES mapping. We derive the L̂ · L̂ strength so thatthe cranking moment of inertia for the boson intrinsic state [14], calculated at
the energy minimum, is equated to the cranking moment of inertia in the HFB
framework at its corresponding minimum in the HFB PES.
To describe the γ-softness in the IBM, it has been shown [15] to be necessary
to consider up to three-body boson Hamiltonian leading to triaxial mean-field
minimum. An illustrative example is presented for 134 Ba in Figure 1, where
both the mapped IBM, which includes specific three-body boson terms, and theHFBCS PESs equally show an almost totally γ-flat structure.
2.3

Benchmarks for basic quadrupole modes

The diagonalization of the mapped IBM Hamiltonian gives description of excitation energies and transition rates. We show in Figure 2(a) the calculated
level energies of the positive-parity states of 146−158 Sm in comparison to theexperimental data [16]. Our calculation predicts, for 148 Sm, the ratio R4/2 =+E(4+1 )/E(21 ) to be close to 2, being characteristic of vibrational nuclei. The

Excitation energy (MeV)

3.0(a) Sm

3.0
Expt.

IBM

2.5

02+

2.0

22+

1.5
1.0
0.5

81+
61+
41++
210.0 84 86 88 90 92 94 96
84 86 88 90 92 94 96
Neutron Number

Excitation energy (MeV)

3.5

2.5
2.0
1.5
1.0
0.5
0.0(b) 134 Ba
6+ 0+
4+0+
3++
4
2+
2+0+

6+ 0+
4+ 0+
3+
4 +2 +2+Expt.

0 + IBM

Figure 2. (Color online) Low-energy spectra for 146−158 Sm (a) and 134 Ba (b) isotopes.

348

Microscopic and Algebraic Theory for Complex Nuclear Shapes and Dynamics
calculated yrast levels become compressed with N , and eventually exhibit atypical rotational band with R4/2 ≈ 3.33, especially for nuclei with N ≥ 90.
An excellent agreement with the experiment is obtained for the rotational yrast
states for heavier isotopes with N ≥ 90. The origin of the overestimation of thenon-yrast energy levels is beyond the scope of the present method and deserves
further investigation. It has been also shown [5, 12] that the IBM parameters
determined by the present method were compatible with those used in previous
phenomenological IBM calculations and derived by the OAI mapping. We show
in Figure 2(b) the low-energy spectrum of 134 Ba, with the typical level structureof a γ-soft nucleus or SO(6) limit [4] of the IBM in a good agreement with thedata [16].
3

Recent Results

Having demonstrated that the DFT-IBM method works well in describing basic
low-energy quadrupole modes – spherical vibrational, strongly deformed and γsoft shapes, we then discuss its recent applications to shape coexistence [17, 18]
and to octupole collective states [19, 20].
3.1

Shape coexistence

The phenomenon of shape coexistence is observed as several 0+ excited statesnear the ground state in specific mass regions. Such low-lying 0+ excited states
can be attributed to the particle-hole excitations in a spherical shell model, andare associated with the different minima in a mean-field picture [21]. Let us
take the 186 Pb nucleus as an illustration. On the left-hand side of Figure 3, theβγ PES calculated within the constrained HFB method, based on the Gogny [22]
EDF with the parametrization D1M [23], exhibits three minima at spherical (β ≈

2.0
1.5

2.0

100.2
β0.3

0.4

0.5

0.3

Mapped IBM40γ (deg)
2.0

0.1

0.1

30

4.5

3.5

β0.2
2.0 1.5
1.0
0.5

1.0

1.5
1.0
0.5

0.1

20

4
4.0 .5
3.5
2.53.0

1.5

0.1

40
γ (deg)
30

50

1.5

β0.2

HFB

0.4

2.5

0.3

0.5

50

4.0

3
2.5 .020
10

1.5
1.00.5

Pb 0.41860.2
β0.3

0.4

0.5

Figure 3. (Color online) Same as for the caption to Figure 1, but for the 186 Pb nucleus.
Gogny-D1M force has been used.

349

K. Nomura

Excitation energy (MeV)

2+Expt.

186

Pb

8+6+

263

460(160)

4510(120)+8+

6200(140)

1D1M−IBM

8++2++384

246+4+0+

2

6140

369++

42

6Obl.

60+

016(2)

002

219 +21+4225 +0Prol.

+0Sph.

Figure 4. (Color online) Experimental [16] and calculated energy spectra and B(E2)transition rates (in Weisskopf units) in the 186 Pb nucleus.

0), oblate (β ≈ 0.2, γ = 60◦ ) and prolate (β ≈ 0.3, γ = 0◦ ) configurations.
Within the IBM, the shape coexistence has been studied by invoking several different IBM Hamiltonians, each corresponding to 2np − 2nh (n = 0, 1, 2, . . .)excitation, and the interactions Ĥmix mixing these Hamiltonians [24]. The
boson model space and the intrinsic state are defined in the enlarged space[(0p − 0h) ⊕ (2p − 2h) ⊕ (4p − 4h) ⊕ · · · ]. For 186 Pb, exhibiting three minima in its HFB PES, we consider up to 4p-4h excitation. Parameters in the total
Hamiltonian Ĥtot = Ĥ0p−0h + Ĥ2p−2h + Ĥ4p−4h + Ĥmix cannot be determined
simultaneously but by taking several steps: associate each unperturbed Hamiltonian Ĥ2np−2nh separately to its corresponding mean-field minimum, and Ĥminis fixed so that barrier heights separating neighboring minima are reproduced(for more details, see [18]).
In Figure 4 we compare the theoretical energy levels, obtained by diagonalizingĤtot in the [(0p − 0h) ⊕ (2p − 2h) ⊕ (4p − 4h)] space, with the data. Our prediction, that three quadrupole collective bands built on the low-energy 0+ states
appear and that two excited 0+ states are composed of either an oblate or prolate
configuration, is consistent with the experimental data and also compares well
the results from previous beyond mean-field calculations on the same nucleus.
The same level of quantitative agreement with experiment has been attained forother Pb isotopes [17] and neutron-deficient Hg isotopes [18].
3.2

Octupole deformations

Next we report the analysis of the quadrupole-octupole correlation where the reflection symmetry (parity) is broken and the negative-parity states emerge [25].
350Microscopic and Algebraic Theory for Complex Nuclear Shapes and Dynamics

Figure 5. (Color online) The DD-PC1 energy surfaces of the isotopes 222−232 Th in the(β2 , β3 ) plane. The color scale varies in steps of 0.2 MeV. The energy difference between
neighboring contours is 1 MeV. Open circles denote the absolute energy minima.

Figure 5 displays the axially symmetric β2 β3 PESs of the 222−232 Th nuclei calculated using the relativistic Hartree-Bogoliubov model with the DD-PC1 parameter set [26]. Already at the mean-field level, one observes a very interesting structural evolution. A near spherical shape is predicted for 222 Th. The
quadrupole deformation becomes more pronounced in 224 Th, and one also notices the development of octupole deformation β3 6= 0. From 224 Th to 226,228 Th
a rather strongly marked octupole minimum is predicted. The deepest octupole
minimum is calculated in 226 Th whereas, starting from 228 Th, the minimum becomes softer in β3 direction. Soft octupole surfaces are obtained for 230,232 Th.
To describe reflection asymmetric shapes and the corresponding negative-parity
states, in addition to the usual positive-parity sd bosons, our IBM model space
includes also the negative-parity f boson [27]. The suitably chosen sdf -IBM
Hamiltonian is determined by mapping β2 β3 DD-PC1 PES onto the equivalent
sdf -boson system [19, 20].
In Figure 6 we display the systematics of the calculated excitation energies of thepositive- and negative-parity yrast states in 220−232 Th in comparison to available
data [16]. Firstly we note that, our calculation quantitatively reproduces the massdependence of the excitation energies of levels that belong to the lowest bands ofpositive and negative parity. The positive-parity states systematically decrease351

K. Nomura

Excitation energy (MeV)

3.0(a) Positive parity

2.5
2.0
1.5 6 +
1.0 4 +
0.5 2 +
0.0

2+
4+
6+
8+
10 +
10 +8+

220 222 224 226 228 230 232
Neutron Number(b) Negative parity1−
3−
5−
7−
9−
9−
7−
5−−
3−
1

220 222 224 226 228 230 232
Neutron Number

Figure 6. (Color online) Excitation energies of low-lying yrast positive- and negativeparity states of 220−232 Th. In each panel lines and symbols denote the theoretical andexperimental [16] values, respectively.

in energy with mass number, reflecting the increase of quadrupole collectivity.
For instance, 220,222 Th exhibit a quadrupole vibrational structure, whereas pronounced ground-state rotational bands are found in 226−232 Th. The levels ofnegative-parity states display a parabolic behaviour centered between 224 Th and226
226
Th. The approximate parabola of the 1−Th,
1 states has a minimum atin which the octupole minimum is most pronounced [cf. Figure 5]. Startingfrom 226 Th the energies of negative-parity states systematically increase and theband becomes more compressed. A rotational-like collective band based on theoctupole vibrational state, i.e., the 1−
1 band-head, develops.
4Conclusions

In this contribution, we have given an overview of the recently developed methodof combining the nuclear DFT with the IBM. The DFT allows a complete meanfield description of low-energy nuclear structure over all masses. The IBM isa well-established theory for a phenomenological description of nuclear collective excitations. To describe nuclear shapes and collective excitations based ona global framework, it has been proposed to build a bridge between the DFT andthe IBM: the Hamiltonian of the IBM is derived by mapping the fermionic onto
bosonic intrinsic states. The proposed method exploits the advantages of both
DFT and IBM: the DFT gives a microscopic input with the IBM, and the IBMthen allows a parameter-free description of nuclear spectroscopy from light toheavy masses, which is much demanding with the DFT calculation. The method
turns out to be valid in reproducing basic quadrupole modes (vibrational, rotational and γ-soft regimes). This points to the first comprehensive way of deriving
the IBM from nucleonic degrees of freedom. The method has the potential to
be extended to complex shape-related phenomena that are of much interest in
nuclear physics, such as shapes coexistence and octupole deformation.
352

Microscopic and Algebraic Theory for Complex Nuclear Shapes and DynamicsAcknowledgments
The author acknowledges supports from the Marie Curie Actions grant within
the Seventh Framework Program of the European Commission under Grant No.
PIEF- GA-2012-327398, and from the Bulgarian National Science Fund underContract No. DFNI-E02/6.
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