1. Nuclear Low-lying Spectrum and Quantum Phase Transition 李志攀 西南大学物理科学与技术学院

2. Introduction Theoretical framework 2 3 3 1 Results and discussion 4 2 Summary and outlook

3. 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 3 0+0+ 2+2+ 4+4+ 6+6+ 8+8+ 3-3- 5-5- 7-7- 9-9- 1-1-

4. 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. Baumann Nature06213(2007) Z N N=20 N=28 N=16 3

5. 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 3

6. 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 ECritical β Potential Order par. Potential Order par. 4 F. Iachello, PRL2004

7. 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 4 Ring96, Vretenar2005, Meng2006 PES AMP+GCM: Niksic2006, Yao2010 5D Collective Hamiltonian based on CEDF

8. Brief Review of the model 5 Construct 5-dimensional Hamiltonian (vib + rot) E(J π ), 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)

9.  Spherical to prolate 1 st 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 2 nd order QPT [Z.P. Li, T. Niksic, D. Vretenar, J. Meng, PRC81, 034316 (2010)] 10 Microscopic Analysis of nuclear QPT

10.  Potential Energy Surfaces (PESs) 12 Discontinuity First order QPT

11.  Potential Energy Surfaces (PESs) 12 along β along γ First order QPT

12.  Spectrum 13... detailed spectroscopy has been reproduced well !! First order QPT

13.  Spectrum  Characteristic features: 13 Sharp increase of R 42 =E(4 1 )/E(2 1 ) and B(E2; 2 1 →0 1 ) in the yrast band X(5) First order QPT

14.  Single-particle levels 14 First order QPT 150 Nd

15. Microscopic analysis of Order parameters  Finite size effect (nuclei as mesoscopic systems)  Microscopic signatures (order parameter) 15  In finite systems, the discontinuities are smoothed out  1 st order 2 nd order; 2 nd order crossover F. Iachello, PRL2004 based on IBM 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)

16.  Microscopic signatures (order parameter) 16 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

17. Second order QPT  Are the remarkable results for 1 st order QPT accidental ?  Can the same EDF describe other types of QPT in different mass regions ? 17 R. Casten, PRL2000 F. Iachello, PRL2000

18. Second order QPT  PESs of Ba isotopes 18

19. Second order QPT  PESs of Xe isotopes 19

20. Second order QPT  Evolution of shape fluctuation: Δβ / 〈 β 〉, Δγ / 〈 γ 〉 20

21. Second order QPT  Spectrum of 134 Ba 21  Microscopic predictions are consistent with data and E(5) for g.s. band  Sequence of 2 2, 3 1, 4 2 : well structure / ~0.3 MeV higher  The order of two excited 0+ states is reversed

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 1 st & 2 nd 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. 22 SummarySummary

23.  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 1 st & 2 nd 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. SummarySummary 22

24.  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 1 st & 2 nd 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. SummarySummary 22

25.  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 1 st & 2 nd 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. SummarySummary 22

26.  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 OutlookOutlook 23

27. 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 26

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29. 6 Collective Hamiltonian

30. 7 Collective Parameter

31. 7

32. 7

33.  Numerical solution of 5D Hamiltonian 8 Numerical Details Spherical E β NSpin (ћ)Energy (ћω) 002.52.500001 123.53.500001 204.54.500001 24.54.500001 44.54.500001 InitialFinalB(E2) (10 -2 t 2 ) 21+21+ 01+01+ 0.50.500006 41+41+ 21+21+ 1.01.000011 61+61+ 41+41+ 1.51.500020 02+02+ 21+21+ 1.01.000011 23+23+ 02+02+ 0.70.700008 Both the excited energy and BE2 are perfectly reproduced

34.  Convergence of the collective parameters 9 Numerical Details For the medium heavy nuclei, N=14 can give convergent result

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36. Shape fluctuation 36