Angular-Momentum Projected Potential Energy Angular-Momentum Projected Potential Energy Surfaces Based on A Combined Method and Tensor Surfaces Based on A Combined Method and Tensor

Force Effects within A Skyrme-HFB Approach Force Effects within A Skyrme-HFB Approach

Jianzhong GuJianzhong Gu

(China Institute of Atomic Energy, Beijing, China)(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 Angular-Momentum Projected Potential Energy Surfaces Based on A Combined MethodCombined Method was 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 byTensor 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).

OutlineThe combined methodand its justification80,82,84Zr Hg isotopesTensor Force Effects

Summary

The combined method and its justificationThere 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 Coulomb interaction of protons which is indispensable for the potential energy surface(PES). To remedy this shortcoming of the PSM and compute the AMPPESs we combine the PSM with 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 the pairing correlations by a separable Gogny D1S force (QCRHB-NL3+separable-Gogny-D1S-force-theory), We first calculate the ground-state PES based on the QCRHB-NL3+separable-Gogny-D1S-force-theory. Then we calculate the PES with a given angular momentum in the framework of the PSM. Finally, the energy difference between the PSM calculated PES with a non-zero angular momentum and that with zero spin is added to the ground-state PES, and a new PES is then formed, which, roughly speaking, has a given angular momentum. Certainly, since angular momentum projection has not yet been performed to the ground-state PES, anything added to the top of it is also unprojected. Those new PESs together with the ground-state PES constitute a group of the PESs with (approximate) given angular momenta. We would say that the ground-state PES serves as a kind of the band head of the PES group.

We would furthermore give some justifications for our combinedWe would furthermore give some justifications for our combined method ofmethod of calculation of the AMPPESs as follows. calculation of the AMPPESs as follows. (a) For a great(a) For a great variety of many-body systems (including the nucleus), it is povariety of many-body systems (including the nucleus), it is possiblessible to describe the excitation spectra in terms of elementary modes ofto describe the excitation spectra in terms of elementary modes of exexcitationcitation representing the different, approximately independent,representing the different, approximately independent, fluctuationfluctuationss aboutabout equilibriumequilibrium(( A. Bohr and B. R. Mottelson, Nuclear Structure (World S A. Bohr and B. R. Mottelson, Nuclear Structure (World Scientific, Singapore, 1998), Vol. 2cientific, Singapore, 1998), Vol. 2)). This implies a. This implies a separation of scale betweeseparation of scale between the excitations of many-body systems andn the excitations of many-body systems and their ground-state energies and their ground-state energies and justifies therefore our combinedjustifies therefore our combined method where nuclear ground states are trmethod where nuclear ground states are treated with the RHB, andeated with the RHB, and nuclear excitations are described by the PSM.nuclear excitations are described by the PSM. There is a separation of energy scales.(b) The Nilsson + BCS(b) The Nilsson + BCS quasiparticle states in the PSM are different from the Rquasiparticle states in the PSM are different from the RHBHB quasiparticle states, which is also justified by the separation ofquasiparticle states, which is also justified by the separation of scale. In fscale. In fact, the former only serves as a basis for the PSM.act, the former only serves as a basis for the PSM.(c)(c) The Hamiltonian used in the PSM is rather schematic forThe Hamiltonian used in the PSM is rather schematic for nuclear excitatinuclear excitations, ons, however, it takes into account the mosthowever, it takes into account the most important long-range correlatimportant long-range correlations (the quadrupole-quadrupoleions (the quadrupole-quadrupole correlation) and the most important shorcorrelation) and the most important short-range correlations (thet-range correlations (the pairing forces)pairing forces) (see, for example, (see, for example, P. Ring and P. SchP. Ring and P. Schuck, The nuclear many body problem, Spinger-Verlag (1980)uck, The nuclear many body problem, Spinger-Verlag (1980))). In this sense, t. In this sense, the PSM is ahe PSM is a shell-modelshell-model like approach.like approach.

Main Points of the Angular Momentum Projected Shell ModelMain Points of the Angular Momentum Projected Shell Model

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.

80,82,84Zr lie 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.

Based on Based on our combinedour combined method ofmethod of calculaticalculation of the AMPPESson of the AMPPESs,for,for 80,82,84Zr, we investigate we investigate their structure and structural evolution their structure and structural evolution with deformation and spinwith deformation and spin.

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

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

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 Decay out in 8484Zr Zr C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R)C. J. Chiara et al., Phys. Rev. C 73 (2006) 021301(R)..Shape transitions occur in Shape transitions occur in 80,8480,84 Zr, which are driven by the 1g9/2 orbital. Zr, which are driven by the 1g9/2 orbital.

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.

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 covariant energy density functional with point-coupling interaction, arXiv:1002.1789v1 [nucl-th]; Phys. Rev. C 82 (2010) 054319.

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), Phys. Rev. C 82(2010)024309.

Different band heads

Bare interactionsThe 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 by exchange of mesons through an effective Lagrangian (EFT)

Gogny interaction

Point coupling

From Fig.2 one can see that the AMP-QCPC-PK1+BCS approach yields the equilibrium shapes which 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 the experimental data. So it is not a good approach to study the ground-state PESs.

The The experimental experimental data: a strongly data: a strongly prolate shape prolate shape >>+0.4 for +0.4 for 8080Zr , Zr , approx 0.3 forapprox 0.3 for8282Zr and Zr and approx 0.2 approx 0.2 forfor 84 84Zr.Zr.

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.

184Hg 186Hg 188Hg 190Hg 192Hg 194Hg 196Hg 198Hg

HFB 0.271 -0.165 -0.136 -0.143 -0.14 -0.106 -0.132 -0.12

0

FRDM -0.125 -0.125 -0.125 -0.125 -0.125 -0.125 -0.117 -0.11

7

RHB 0.286 0.284 0.283 -0.139 -0.137 -0.104 -0.101 -0.09

6

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 ModelHFB: HFB-full-Gogny-D1SRHB: QCRHB-NL3+separable-Gogny-D1S-force-theory

Jianzhong Gu, Bangbao Peng, Wenhua Zou and Shuifa Shen,Nucl. Phys. A 834 (2010) 87c.

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

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.

Table 1 Tunneling width Γtunn (in units of eV)

JJ 188188HgHg 190190HgHg 192192HgHg 194194HgHg 196196HgHg

0022446688101012121414161618182020

753.2753.2684. 3684. 3607.1607.1486.486.397.397.207.207.180.180.

00000000

16.716.714.314.312.912.911.111.17.07.02.62.6

6.8E-56.8E-51.7E-71.7E-72.1E-92.1E-9

6.7E-116.7E-112.4E-122.4E-12

47.247.246.546.531.031.030.030.0

15.4015.402.02.0

6.6E-46.6E-45.7E-45.7E-49.1E-79.1E-72.3E-82.3E-8

4.6 E-114.6 E-11

3.23.22.42.41.91.91.11.1

0.480.480.330.330.130.13

2.4E-32.4E-31.5E-61.5E-63.8E-83.8E-81.8E-91.8E-9

0.740.740.580.580.440.440.270.270.120.12

0.0750.0750.0270.0270.0110.011

0.00380.00388.6E-68.6E-61.4E-81.4E-8

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., R. Kruecken et al., Phys. Rev. C 64 (2001) 064316). The sudden decay out can Phys. Rev. C 64 (2001) 064316). The sudden decay out can be understood more clearly.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.

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.

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).

Tensor Force Effects within A Skyrme-HFB Tensor Force Effects within A Skyrme-HFB Approach Approach (Only for Spherical Systems)(Only for Spherical Systems)

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

0 2 4 60.00

0.02

0.04

0.06

0.08

0.10

0 2 4 6 8

p (fm

-3)

r (fm)

without tensor

46Ar

T11 T12 T14 T16 T22 T32 T43 T44 SLy5+T SLy5+T

w

with tensor

-56

-48

-40

-32

-24

-16

-8

SLy5+Tw

E (M

eV)

1s1/2

1p3/2

1p1/2

1d5/2

1d3/2

2s1/2

1f7/21f

7/2

2s1/2

1d3/2

1d5/2

1p1/2

1p3/2

1s1/2

SLy5

22

8 8

20 20

46Ar

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.

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).

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

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

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 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 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.

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!

(d) 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.