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From Nucleon Structure to Nuclear Structure and Compact Astrophysical Objects

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Angular-Momentum Projected Potential Energy Surfaces Based on A Combined Method and Tensor Force Effects within A Skyrme-HFB Approach Jianzhong Gu(China Institute of Atomic Energy, Beijing, China)

From Nucleon Structure to Nuclear Structure and Compact Astrophysical Objects

June 22, 2012, KITPC/ITP-CAS, Beijing

Angular-Momentum Projected Potential Energy Surfaces Based on A Combined Methodwas done in collaboration with

Bangbao Peng (CIAE),

Yuan Tian (CIAE),

Wenhua Zou (CIAE),

Jiangming Yao (Southwest U),

Shuifa Shen (East China Institute of Technology),

Zhongyu Ma (CIAE).

Tensor Force Effects within A Skyrme-HFB Approach was done by

Yanzhao Wang (CIAE),

Jianmin Dong (IMP, Lanzhou),

Wei Zuo (IMP,Lanzhou),

Jianzhong Gu (CIAE),

Xizhen Zhang (CIAE).

Outline

The combined method

and its justification

80,82,84Zr

Hg isotopes

Tensor Force Effects

Summary

The combined methodand its justification

There are many nuclear models and theories, their connections are not so clear. Collecting their merits together one may better understand the nuclear many-body problem.

In order to study nuclear equilibrium shapes, shape coexistence, shape transitions and decay out of super-deformed bands we develop a method to calculate nuclear potential energy surfaces (binding energies) by combing a model to describe nuclear ground state properties with another model to describe nuclear excitations . We may Combine the projected shell model with HFB or RHB.

The angular momentum projected potential energy

surfaces ( AMPPESs) method

Let us first Combine angular momentum projected shell model (PSM) with RHB.

The Hamiltonian of the PSM does not contain the Coulombinteraction

of protons which isindispensable for the potential energy surface

(PES). To remedy thisshortcoming of the PSM and compute the AMPPESs we combine the PSMwith the QCRHB-NL3+separable-Gogny-D1S-force-theory.

The quadrupole constrained RHB theory, in which the relativistic mean-field (RMF) Lagrangian is described by the NL3 effective interaction and thepairing correlations by a separable Gogny D1S force (QCRHB-NL3+separable-Gogny-D1S-force-theory),

We first calculate theground-state PES based on theQCRHB-NL3

+separable-Gogny-D1S-force-theory. Then we calculate thePES with a given angular momentum in the framework of the PSM.Finally, the energy difference between the PSM calculated PES with anon-zero angular momentum and that with zero spin is added to theground-state PES, and a new PES is then formed, which, roughlyspeaking, has a given angular momentum.

Certainly, since angularmomentum projection has not yet been performed to the ground-statePES, anything added to the top of it is also unprojected. Those newPESs together with the ground-state PES constitute a group of thePESs with (approximate) given angular momenta. We would say that theground-state PES serves as a kind of the band head of the PES group.

We would furthermore give some justifications for our combinedmethod ofcalculation of the AMPPESs as follows.

(a) For a greatvariety of many-body systems (including the nucleus), it is possible to describe the excitation spectra in terms of elementary modes ofexcitation representing the different, approximately independent,fluctuationsaboutequilibrium( A. Bohr and B. R. Mottelson, Nuclear Structure (World Scientific, Singapore, 1998), Vol. 2). This implies aseparation of scale between the excitations of many-body systems andtheir ground-state energies and justifies therefore our combinedmethod where nuclear ground states are treated with the RHB, andnuclear excitations are described by the PSM.

There is a separation of energy scales.

(b) The Nilsson + BCSquasiparticle states in the PSM are different from the RHB quasiparticle states, which is also justified by the separation ofscale. In fact, the former only serves as a basis for the PSM.

(c)The Hamiltonian used in the PSM is rather schematic fornuclear excitations, however, it takes into account the mostimportant long-range correlations (the quadrupole-quadrupolecorrelation) and the most important short-range correlations (thepairing forces) (see, for example, P. Ring and P. Schuck, The nuclear many body problem, Spinger-Verlag (1980)). In this sense, the PSM is ashell-modellike approach.

Main Points of the Angular Momentum Projected Shell Model

Based on our combinedmethod ofcalculation of the AMPPESs,for80,82,84Zr,we investigate their structure and structural evolution with deformation and spin.

Near the N=Z line, abundant and exotic nuclear structure due to large parts of protons and neutrons in pfg orbitals, level density high, and a severe competition between single-particle motions and collective motions. and the intruder of the 1g9/2 further complicates the structure and plays an important role in the shape coexistence.

80,82,84Zrlie near the proton drip-line and are very exotic nuclei.

The even-even N=Z waiting point nuclei are very important for nuclear astrophysical rp process.

Recently, experimental study for nuclei with N=Z：Nature 469 (2011) 68.

The QCRHB-NL3+separable-Gogny-D1S-force-theory reproduces experimental data of the equilibrium shapes.

A=80 mass region，shapes, shape coexistence, shape transitions and decay out of the super-deformed bands.

The decay out could be rather fragmented since the energy difference between the SD states and ND (spherical) states is as high as 6-8 MeV for 82Zr and 84Zr nuclei at high spins. Nevertheless, for 80Zr nucleus, there is no decay out of the SD band since the barrier is so thick.

Decay out in 84Zr C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R).

Shape transitions occur in 80,84 Zr, which are driven by the 1g9/2 orbital.

Different band heads

The quadrupole constrained RHB theory, in which the relativistic mean-field (RMF) Lagrangian is described by the NL3 effective interaction and the

pairing correlations by a separable Gogny D1S force (QCRHB-NL3+separable-Gogny-D1S-force-theory), Phys. Rev. C 82(2010)024309.

The quadrupole constrained relativistic mean-field framework with PC-PK1 parameter set. The pairing correlation is considered through a standard BCS method with a density-independent delta pairing force (AMP-QCPC-PK1+BCS approach). P. W. Zhao, Z. P. Li, J. M. Yao et al., New parametrization for the nuclear covariantenergy density functional with point-coupling interaction, arXiv:1002.1789v1[nucl-th]; Phys. Rev. C 82 (2010) 054319.

Hartree-Fock-Bogoliubov (HFB) axial mean field calculations based on the D1S Gogny interaction (HFB-full-Gogny-D1S), Eur. Phys. J. A 33 (2007) 237.

Bare interactions

The most widely used NN potentials are the Paris potential, the Argonne AV18 potential, the CD-Bonn potential and the Nijmegen potentials.

Skyrme interaction

Nucleons are coupled byexchange of mesons through an effectiveLagrangian (EFT)

Gogny interaction

Point coupling

The experimental data: a strongly prolate shape

>

+0.4 for 80Zr , approx 0.3 for

82Zr and approx 0.2

for 84Zr.

From Fig.2 one can seethat the AMP-QCPC-PK1+BCS approach yields the equilibrium shapeswhich are consistent with the experimental data. However, the QCHFB-full-Gogny-D1S approach predicts the spherical equilibriumshapes for the three nuclei, which are inconsistent with theexperimental data. So it is not a good approach to study theground-state PESs.

We recalculated the AMPPESs of 80,82,84Zr nuclei by replacing the QCRHB-NL3+separable-Gogny-D1S-force-theory by the AMP-QCPC-PK1+BCS approach, and found that the two kinds of AMPPESs have a few common features:The strong shape mixing in 82Zr and the decay out of the SD bands in nuclei 82,84Zr although at low spins they are different from each other. The common features imply that the strong shape mixing and the decay out of the SD bands are not so sensitive to the choice of the band heads.

For Hg isotopes, there are rich data on the decay out phenomenon.

Hartree-Fock-Bogoliubov (HFB) axial mean field calculations based on the D1S Gogny interaction (HFB-full-Gogny-D1S), Eur. Phys. J. A 33 (2007) 237.

FRDM: Finite Range Droplet Model

HFB: HFB-full-Gogny-D1S

RHB: QCRHB-NL3+separable-Gogny-D1S-force-theory

Experiments: for A=190 mass region, I=8-12 hbar at decay out points.

Calculations:the barrier gets thin and low at such spins, and the decay out suddenly happens.

Information for the excitation energies and spins at decay out points can be obtained, which is the most wanted for experiments.

Bandheads here are taken from Eur. Phys. J. A 33 (2007) 237. A HFB approach based on the D1S Gogny force.

Jianzhong Gu, Bangbao Peng, Wenhua Zou and Shuifa Shen,

Nucl. Phys. A 834 (2010) 87c.

Table 1 Tunneling width Γtunn (in units of eV)

The tunneling width could be identical to the spreading width, which shares the same order of magnitude as those predicted by the GW model (for instance, R. Kruecken et al., Phys. Rev. C 64 (2001) 064316). The sudden decay out can be understood more clearly.

There is no “one size fits all”nuclear theory. If you think all of the extant models and theories are useful, and if you want to develop a unified theory for the nuclear many-body problem you have to consider the relationship of the extant models and theories, which could be tough. Let us first bring them together, we may learn more and better.

On the one hand one can combine different models or theories and figure out the most degrees of freedom to understand the nuclear many-body problem better and more. On the other hand one should complete indispensable ingredients for nucleon-nucleon interactions.

Tensor Force Effects within A Skyrme-HFB Approach

(Only for Spherical Systems)

The tensor force is a necessary and important component of the nuclear force and had been ignored for a long time. In the framework Skyrme-HFB the tensor force effects on nuclear structure are investigated by YZ Wang, JM Dong et al.

For light nuclei, their structure changes due to the tensor force are significant.

Tensor force is non-central and non-local spin-spin interaction, within the framework of Skyrme-HFB, has the following form:

Yanzhao Wang, Jianzhong Gu, Xizhen Zhang et al., Phys. Rev. C 83,054305 (2011).

The tensor force causes the inversion between the levels of 2s1/2 and 1d3/2 in the nuclei around 46Ar. The inversion results in the proton bubble formation in 46Ar.

Yanzhao Wang, Jianzhong Gu, Xizhen Zhang et al., Chin. Phys. Lett. 28,102101 (2011).

Yanzhao Wang, Jianzhong Gu, Xizhen Zhang et al., Phys. Rev. C 84, 044333 (2011).

The tensor force has significant influence on the pseduo-spin energy splittings and shell correction energy.

Jianmin Dong, Wei Zuo, Jianzhong Gu et al., Phys. Rev. C 84 (2011) 014303.

Add something necessary to the Skyrme interactions, and constrain them using additional criteria.

Pseduo-spin enregy splittings in the cases with and without the tensor force within the Skyrme-HFB framework with the TIJ parameterizations.

By YZ Wang, ZY Li, GL Yu et al.

A classification of the TIJ parameterizations according to the changes of the pseduo-spin energy splittings caused by the tensor force.

By YZ Wang, ZY Li, GL Yu et al.

The single-particle energy differences in the cases with and without the tensor force within the Skyrme-HF framework with the TIJ parameterizations.

By ZY Li , YZ Wang, GL Yu et al.

The single-particle energy differences in the cases with and without the tensor force within the Skyrme-HF framework with the TIJ parameterizations.

By ZY Li , YZ Wang, GL Yu et al.

The shell gaps in the cases with and without the tensor force within the Skyrme-HF framework with the TIJ parameterizations.

By ZY Li , YZ Wang, GL Yu et al.

- Summary
- (a) We proposed a new method to compute angular momentum projected nuclear
- potential energy surfaces.
- (b) The equilibrium shapes, shape coexistence, shape transitions and decay out of
- super-deformed bands for 80,82,84 Zr nuclei were studied with the two different
- band heads. We found that for low spins, the AMPPESs with the two different
- band heads are quite different, in the case of high spins and large
- deformations they nevertheless share so much in common. And both of them
- predict the strong shape mixing in 82Zr.
- (c) The decay out for Hg isotopes is isospin dependent, the barrier gets thin and
- low at low spins, and the decay out suddenly happens around the first back-
- bending which is due to the pair breaking, not the degree of the chaoticity.
- A challenge : describe the evolution of nuclear structure in a wide range of
- deformation!
- Significant influence of the tensor force on the structure of light nuclei and their evolution has been found and the effect of the tensor force on pseudo-spin energy splittings and shell correction is evident.

- (e) The proton bubble in 46Ar has been found which results from the inversion of the
- proton single particle levels.
- (f) One may constrain extant the Skyrme interactions with the tensor force by using
- additional criteria.