Nuclear Low-lying Spectrum and Quantum Phase Transition

李志攀西南大学物理科学与技术学院

第十三届全国核结构研讨会暨第九次全国“ 核结构与量子力学”专题讨论会

Introduction

Theoretical framework2

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1

Results and discussion

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2

Outline

Summary and outlook

Nuclear Low-lying Spectrum Nuclear low-lying spectrum is an important physical

quantity that can reveal rich structure information of atomic nuclei

Shape and shape transition

30+2+4+6+8+

3-5-7-9-

1-

Nuclear Low-lying Spectrum Nuclear low-lying spectrum is an important physics

quantity that can reveal rich structure information of atomic nuclei

Shape and shape transition Evolution of the shell structure

T. BaumannNature06213(2007)

Z

N

N=20

N=28

N=163

Nuclear Low-lying Spectrum Nuclear low-lying spectrum is an important physics

quantity that can reveal rich structure information of atomic nuclei

Shape and shape transition Evolution of the shell structure Evidence for pairing correlation

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Quantum Phase Transition in finite system Quantum Phase Transition (QPT) : abrupt change of

ground-state properties induced by variation of a non-thermal control parameter at zero temperature.

In atomic nuclei:First and second order QPT can occur between systems characterized by different ground-state shapes.

Control Par. Number of nucleons

Two approaches to study QPT Method of Landau based on potentials (not observables) Direct computation of order parameters (integer con. par.)

Combine both approaches in a self-consistent microscopic framework

Spherical

Deformed

E Critical

β

Potential Order par.

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F. Iachello, PRL2004

Covariant Energy Density Functional (CEDF) CEDF: nuclear structure over almost the whole nuclide

chart

Scalar and vector fields: nuclear saturation properties Spin-orbit splitting Origin of the pseudo-spin symmetry Spin symmetry in anti-nucleon spectrum ……

Spectrum: beyond the mean-field approximation

Restoration of broken symmetry, e.g. rotational Mixing of different shape configurations

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Ring96, Vretenar2005, Meng2006

PES

AMP+GCM: Niksic2006, Yao2010

5D Collective Hamiltonian based on CEDF

Brief Review of the model

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Construct 5-dimensional Hamiltonian(vib + rot)

E(Jπ), BE2 …

Cal. Exp.

3D covariant Density Functional

ph + pp

Coll. Potential

Moments of inertia

Mass parameters

Diagonalize:Nuclear spectroscopy

T. Niksic, Z. P. Li, D. Vretenar, L. Prochniak, J. Meng, and P. Ring 79, 034303 (2009)

Spherical to prolate 1st order QPT [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, G.A. Lalazissis, P. Ring, PRC79, 054301(2009)]

Analysis of order parameter [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, PRC80, 061301(R) (2009)]

Spherical to γ-unstable 2nd order QPT [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, PRC81, 034316 (2010)]

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Microscopic Analysis of nuclear QPT

Potential Energy Surfaces (PESs)

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Discontinuity

First order QPT

Potential Energy Surfaces (PESs)

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along β along γ

First order QPT

Spectrum

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... detailed spectroscopy has been reproduced well !!

First order QPT

Spectrum

Characteristic features:

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Sharp increase of R42=E(41)/E(21) and B(E2; 21→01) in the yrast band

X(5)

First order QPT

Single-particle levels

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First order QPT

150Nd

Microscopic analysis of Order parameters Finite size effect (nuclei as mesoscopic systems)

Microscopic signatures (order parameter)

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In finite systems, the discontinuities are smoothed out 1st order 2nd order; 2nd order crossover

F. Iachello, PRL2004 based on IBM

1. Isotope shift & isomer shift

2. Sharp peak at N~90 in (a)

3. Abrupt decrease; change sign in (b)

Microscopic signatures (order parameter)

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Conclusion: even though the control parameter is finite number of nucleons, the phase transition does not appear to be significantly smoothed out by the finiteness of the nuclear system.

Microscopic analysis of Order parameters

Second order QPT

Are the remarkable results for 1st order QPT accidental ? Can the same EDF describe other types of QPT in different

mass regions ?

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R. Casten, PRL2000F. Iachello, PRL2000

Second order QPT PESs of Ba isotopes

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Second order QPT PESs of Xe isotopes

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Second order QPT Evolution of shape fluctuation: Δβ/ 〈 β 〉 , Δγ/ 〈 γ 〉

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Second order QPT Spectrum of 134Ba

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Microscopic predictions are consistent with data and E(5) for g.s. band Sequence of 22, 31, 42 : well structure / ~0.3 MeV higher The order of two excited 0+ states is reversed

5D Collective Hamiltonian based on CEDF has been constructed

Microscopic analysis of nuclear QPT

PESs display clear shape transitions The spectrum and characteristic features have

been reproduced well for both 1st & 2nd order QPT The microscopic signatures have shown that the

phase transition does not appear to be significantly smoothed out by the finiteness of nuclear system.

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Summary

5D Collective Hamiltonian based on CEDF has been constructed

Microscopic analysis of nuclear QPT

PESs display clear shape transitions The spectrum and characteristic features have

been reproduced well for both 1st & 2nd order QPT The microscopic signatures have shown that the

phase transition does not appear to be significantly smoothed out by the finiteness of nuclear system.

Summary

22

5D Collective Hamiltonian based on CEDF has been constructed

Microscopic analysis of nuclear QPT

PESs display clear shape transitions The spectrum and characteristic features have

been reproduced well for both 1st & 2nd order QPT The microscopic signatures have shown that the

phase transition does not appear to be significantly smoothed out by the finiteness of nuclear system.

Summary

22

5D Collective Hamiltonian based on CEDF has been constructed

Microscopic analysis of nuclear QPT

PESs display clear shape transitions The spectrum and characteristic features have

been reproduced well for both 1st & 2nd order QPT The microscopic signatures have shown that the

phase transition does not appear to be significantly smoothed out by the finiteness of nuclear system.

Summary

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Further application: Systematic investigation of nuclear QPT Shape coexistence, e.g. Kr & Pb ……

Development of the model:

Cranking CEDF: Thouless-Valatin moment of inertia Constraint on collective P: mass parameters Coupling between nuclear shape oscillations and

pairing vibrations

Outlook

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孟杰教授 , 张双全博士及北大 JCNP 全体成员 （北京大学）赵恩广研究员 , 周善贵研究员 （中科院理论物理研究所）龙文辉教授 （兰州大学）尧江明教授 （西南大学）孙保华博士 （北京航空航天大学）彭婧博士 （北京师范大学）王守宇博士，亓斌博士 （山东大学）张炜博士 （河南理工大学）Prof. D.Vretenar ， Dr. T.Niksic ， Prof. N.Paar （ Zagreb, Croatia ）Prof. P. Ring (TUM, Germany)Prof. J.Libert, Prof. E.Khan, Prof. N. Van Giai （ IPN-Orsay, France ）Prof. G. Lalazissis (Thessaloniki, Greece)Prof. G. Hillhouse (Stellenbosch, South Africa)Prof. L. Prochniak (Lublin, Poland)Prof. L. N. Savushkin (St. Petersburg, Russia)

Acknowledgments

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Thank you For your attention

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Collective Hamiltonian

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Collective Parameter

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Collective Parameter

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Collective Parameter

Numerical solution of 5D Hamiltonian

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Numerical Details

SphericalE

β

N Spin (ћ) Energy (ћω)

0 0 2.5 2.500001

1 2 3.5 3.500001

2 0 4.5 4.500001

2 4.5 4.500001

4 4.5 4.500001

Initial Final B(E2) (10-2t2)

21+ 01

+ 0.5 0.500006

41+ 21

+ 1.0 1.000011

61+ 41

+ 1.5 1.500020

02+ 21

+ 1.0 1.000011

23+ 02

+ 0.7 0.700008

Both the excited energy and BE2

are perfectly reproduced

Convergence of the collective parameters

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Numerical Details

For the medium heavy nuclei, N=14 can give

convergent result

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Shape fluctuation