1. 原子核配对壳模型的相关研究 Yanan Luo( 罗延安 ), Lei Li( 李磊 ) School of Physics, Nankai University, Tianjin Yu Zhang( 张宇 ), Feng Pan( 潘峰 ) Department of Physics, Liaoning Normal University, Dalian J. P. Draayer Department of Physics and Astronomy, LSU, BTR Frontiers in Hadron and Nuclear Physics, 2015.01.10 Beijing Huairou

2. Outline I. Introduction II. Nucleon-pair shell model(NPSM) III. Work about the NPSM VI. Summary

3. I. Introduction Vibrational spectrum Rotational spectrum

4. Pairing model Interacting boson model Fermion dynamical symmetry model Rigid rotor model Shell model ……….

5. How to describe the collectivity in term of shell model? The problem in the shell model The Model space is HUGE! 10 9 for modern computers 10 14-18 for mediam and heavy nuclei and modern computers fail for these nuclei! Truncation of the shell model space must be found!

6. The success of the IBM showes that S and D fermion pairs play important role S and D pairs mapping to s and d boson!

7. II. Nucleon-pair shell model The Hamiltonian

8. The model space

9. How to calculate …….

10. Commutator between coupled operators

12. Overlap between two N-pair states

13. The advantages of the NPSM: 1. it accommodates various truncation, ranging from the truncation to only the S subspace, the S-D subspace, up to the full shell model space 2. it is flexible enough to include the broken pair approximation, the pseudo SU(2) or the favored pair model and the fermion dynamical symmetry model as its special cases Because of the success of the IBM, the model space is truncated to the S- and D-pair subspace

14. III. Work about the NPSM Fitting the experimental data Limiting cases of the IBM Nuclear shape phase transitional pattern …….

15. The Hamiltonian used in fitting the experimental data

24. Summary Nucelon-pair shell model truncated to the SD-pair subspace is reasonable! IBM has sound shell model foundation! Thank you very much for your attention!

25. I. Nuclear Shape-Phases and Shape-Phase Transition Modes of nuclear collective motion and the symmetries  Shape of Nucleus  Sphere  Deforemation quadrupole octupole hexadecupole  Modes of Nuclear Collective Motion vibration axial rotation ( prolate, oblate) prolate  -soft rotation triaxial rotation From Prof. Yuxin Liu

26.  Correspondence between collective motion and symmetry Vibration U(5) Axial Rotation SU(3) ( SU*(3))  γ-soft rotation O(6)  Shape Phase Transition and the States at the Critical Points of the Phase Transitions Vibration – γ-soft Rotation E(5) Vibration – Prolate Axial Rotation X(5) Prolate – Oblate Axial Rotations Y(5)

27. Iachello, PRL 91, 132502 ( ’ 03)

28. Nuclear shape-phase and phase transition in IBM-2 M. A. Caprio and F. Iachello ， Phys. Rev. Lett. 93, 242502 (2004)

29. II. Status of the research on nuclear shape phase transition Even-even nuclei, Odd-even nuclei, Odd-odd nuclei 1.IBM, IBFM, IBFFM 2.Geometric Collective models 3. Fermionic approachs FDSM, shell model, HFB

30. The most useful observables

31. Two neutron seperation energies E0 transition Isotopes shifts Two neutron transfer cross sections ……………….

32. III. SD-pair shell model(SDPSM) Because of the success of the IBM, the full space was truncated into SD-pair subspace, r=0, 2 The Hamiltonian can be diagonolized directly in the Fermion space!

33. IV. Nuclear shape phases in SDPSM

34. Rotational spectrum

36. V. Nuclear shape phase transition in SDPSM

37. Vibration to rotational transitional pattern

39. Boson Mapping (Dyson) Liu YX, et. al., Phys. Lett. B688,298(2010)

40. The Hamiltonian used in the SD-pair shell model 50-82 shell

41. Quadrupolel-quadrupole interactional strength kappa E G 0 =0.1MeV, G 2 =0, k=0~0.05MeV/r 4 0

42. SDPSM results DBM results

43. Summary The nuclear shape phases can be produced very well in the SDPSM The nulcear shape phase transitional pattern as in the IBM can be produced in the SDPSM The properties of the critical symmetry can be produced in the SDPSM

44. Thanks !

45. Spectrum of E(5) Spectrum of X(5) Spectrum of Y(5) Symmetry Symmetry Symmetry ( Iachello, PRL 85, 3580 (2000) ) ( Iachello, PRL 87, 052501(2001)) ( Iachello, PRL 91, 132502(2003)) 134 Ba, 108 Pd, 130 Xe, ··· 152 Sm, 154 Gd, 156 Dy, 150 Nd, ··· 166 Er, 168 Er, ··· ( Casten, PRL 85, 3584 (2000); ( Casten, PRL 87, 052503 (2001); (PRC 68, 024307 (2003); … Ginocchio, PRL 90, 212501 (2003) ; Capirio, PRC 66, 054310 (2002); Liu, PRC 65, 057301 (2002), … ) Tonev, PRC 69, 034334 (2004); … )

46. Characteristic of Evolution of Energy Spectrum for the Transition from U(5) to SU(3) Through X(5) SU(3) (Rotation) X(5) U(5) (Vibration) ( Pan, Draayer, et. al., PLB 576, 297 (2003) )

47. Summary for the shape-phase transition in the U(5)-SU(3) transitional region Harmonic Vibrator Soft Liquid- drop Rigid Rotor

48. Extension: Unified description of the shape phase structure and phase transition of odd-A nuclei F. Iachello, Phys. Rev. Lett. 95, 052503 (2005); Result of U(6,4) model 135 Ba, E(5/4) Symmetry, M.S. Fetea, et al., Phys. Rev. C 73, 051301 (2005)